SOtTHERK  CONFEDERACY 


ARITHMETIC, 


Coiuiuoii  Schools  and  .Academies, 


WITH    A 


ii 


II  i 


PKAOTiCAL    SYSmi-   OK    HOOK-KEEPING 

l>i       SINGI^i:      ENTRY. 


Bi  Rev.  CHARhKS  Jv  KKVKl.CTT,  A.M., 

AUTHOR    OF    THK        KTHKBN    CON^KDHUArV     CLASS    READBFd. 
v;tC.  .    V.1 1". 


lit 


''I'' 


A  IT  G  I    S  r  A  ,      G  A .  : 

POINTED    AND    I'UrMSinF.P    HX.   J.    ^     rVTBBSON    *    CO. 

1861. 


COL.  GEORGE  WASHINGTON  FLOWERS 
MEMORIAL  COLLECTION 


DUKE  UNIVERSITY  LIBRARY 
DURHAM.  N.  C. 


PRESENTED  BY 
W.  W.  FLOWERS 


u  t^ 


THE 


SOUTHERN  CONFEDERACY 


ARITHMETIC, 


FOR 


Common  Schools  and  Academies, 


WITH    A 


PRACTICAL    SYSTEM    OF    BOOK-KEEPING 


BY      SINGLE      ENTRY. 


By  Rev.  CHARLES  E.  LEVERETT,  A.M., 

AUTHOR    OF    TUB    SOUTUKUN    COXKEDKRACY    CLASS    READERS, 
ETC.,     ETC. 


A  U  G  U  S  TA,      G  A.: 

PRINTED    AND    PUBLISHED    BY    J.    T.    PATERSON    <t    CO. 

1864. 


Eutered  according  to  Act  of  Congress  in  the  year  1S64,  by  . 

J.T.PATERSON&CO., 

in  the  Clerk's  Office  of  the  District  Court  of  the  Confederate  States  for  the 
Northern  District  of  Georgia. 


FHEF  A.  C  E. 


A  FEW  years  sinee^  when,  for  reasons  unnecessary  to  be 
repeated  here,  it  was  "thought  advisable  that  the  South  should 
have  the  preparation  of  her  own  educational  literature,  the  author 
was  asked  to  aid  in  the  eaterprisc  conducting  to  that  end.  At 
first,  as  not  being  exactly  in  the  line  of  his  professional  service, 
he  declined ;  but,  after  due  deliberation  and  consultation,  he 
assented,  and  his  earliest  contribution  is  tTie  volume  which  this 
preface  introduces.  In  offering  this  he  begs  to  say,  that  while 
a  loag  scholastic  experience  has  given  him  peculiar  facilities  for 
his  work,  his  deep  interest  in  the  proper  training  of  the  young 
has  materially  aided  in  what,  he  trusts,  will  tend  to  a  good 
result.  Nor  does  he  now  hesitate  to  aflfirm  that  it  will  be  a 
source  of  the  deepest  gratification  to  him,  if,  througli  his  humble 
agency,  any  text  book  of  the  section,  which  has  developed  the. 
most  malignaht  principles  and  exhibited  the  crudest  practice, 
should    be   forever   excluded    from   a   Southern,  school. 

In  the  preparation  of  this  volume,  thc^  author  has  .been 
guided  by  what  he  understood  had  long  been  needed — a  simple 
and  comprehensive  work — one  that  could  be  studied  both  wi,th 
proQt  and  pleasure.'  He  hopes  to  have  reached,  at  least  partly, 
what  seemed  to  him  desirable.  The  books,  and  the  most  popnkr 
niies,  which  the  Northern  market  has  in  immense  numbers 
(■xported  to  the  South,  have  very  glaring  defects,  especially  in 
the    rittoni-     t"-'    ^>ach    too    little,    and    ngain    too    nnu-l)        Tb^ 


VI  PREFACE. 

consequence  lias  been  embarrassment  to  tbe  common  order  of 
minds,  and  a  tendency  to  make  a  most  pleasing  science  an 
occult  study,  and  a  useful  and  necessary  bvauc-b  of  pcience  a 
distasteful  task.  In  tliis  way,  that  course,  vrithout  -wliicb'  no 
education  is  complete,  eitl\er  in  its  relation  to  the  sciences 
connected  with  the  reasoning  powers,  or  in  its  practical 
addresses  in  commercial  engagements,  and  in  the  daily  appli' 
cations  of  domestic  finance,  is  made  to  sustain  a  most  depre- 
ciated  stand. 

It  has  been  the  author's  aim  to  avoid  prolixity.  To 
acquire  numerical  skill,  a  pupil  should  not  become  vreary  of 
his  work.  Arithmetics,  generally,  because  too  long,  are  tire- 
some, and  the  end — which  the  scholar  always  has  in  view — 
appears —    ■ 

"Tho   weary   wild,    that   lengthens   an.    v>'e  go." 

This  book  is  purposely  short,  and,  as  for  the  most  part,  Hhe 
secret  of  scholarly  success  is  the  culture  from  the  review — 
the  nail  to  fasten  science  to  the  mind — it  should  be  repeat- 
edly  used  before  the  higher  volume  of  the  series  is  taken  up. 

It  has  been  an  additional .  aim  to  give,  at  each  progressive 
Step,  clear  explanations.  In  these,  when  it  could  be  done, 
the  rule  for  operations  is  introduced,  and  by  this  plan  has 
been  avoided  what  to  young-  pupils  is  a  very  mystifying  and 
objectionable  paj-t  *of  Arithmetical  study— the  dry  formula  of 
directions.'  In  regard  to  the  explanations,  much  must  be  left 
to  the  teacher,  and  he  is  an  unintelligent  one  who  does  not 
make  plain,  by  illustrations,  certain  truths,  which  no  Arith- 
metic, unless  it  be  expanded  to  an  unreasonable  length,  could 
communicate.  In  this  connection,  it  may  be  well  to  say  that 
it  would  have  been  perfectly  easy  to  have  introduced  a  very 
frequent     analysis,     and    with    it    have    blended    questions,    after 


rUEFACE.  Vll 

the  manner  of  some  Aritlimcticiaus,  in  order  to  cDnibme  tho 
mental  and  written  dcpavtnaents;  but,  after  trying  all  forma, 
his  experience  has  held  it  preferable  to  keep-  tbcni  separate. 
When  too  many  things  arc  presented  together  there  is  con- 
fusion, and  the  contemplated  object  is  not  secured.  For  tho 
encouragement  of  tbe  pupil,  even  if  his  be  not  a  mind,  strictly 
speaking,  mathematically  framed,  it  mny  be  •  said,  if  he  have 
well  studied  .  the  Intellectual  volume  of  this  series,  ho  can 
hardly  fail,  by  the  time  he  has  completed  this,  with  its  review, 
to  comprehend  to  a  fair  degree  what  Arithmetic  has  been 
defined,  the  art  of  numbers,  and  the  science  or  knowledge  of 
computation. 

The  marginal  notes,  it  is  thought,  will  prove  highly  useful 
and  acceptable  as  a  ready  index  of  subjects  :ind  a  form,  for 
questions,  without  the  formality  of  their  shape.  This  new 
feature  has  received  from  intelligent  teachers  very  gratifying 
approbation.  The  divisions  into  Introductory,  Higher,  Frac-r 
tional.  Proportional,  Commercial,  lladical  and  I'.iiscellancous 
Arithmetic,  it  is  hoped,  will  be  a  help  both  to  the  scholar 
and  instructor.  For  the  convenience  of  the  latter,  a  key, 
containing   answers,    will   accompany   the   volume. 

Columbia,  S.  C,  Feb.,  18G4. 


216341 


COISTTEILTTS. 


INTRODUCTORY      ARITHMETIC. 

Page. 

Definitions      ........  i 

notatiok                 ..--...  2 

Numeration                  -■-•--  3 
Addition                               -            -            -            -            -            .7 

Subtraction                 -            -            -            -            -            -            -  1 7 

AfuLTUM.ICATION                            -                  .                  .                  .                  .                  .  23 

DrviSTON           -            -            -            -            -            -            - '           -  32 

Miscellaneous  Examples             -            -            -            .            .  kj 

Reddction                     ■'-----  42 

Tables  and  Examples      ■---..  43 

Miscei^laneocs  Tables            -  .         -            -            -            .            -  59 

FoREiuN  Coins        --■-...  59 

The  Meascri:ment  of  Corn,  Pkas,  Potatoes,  in  Bulk        -  CO 

Reddction  of  Lower  to  Hu;in:i;  N'amkh             -            -            .  ^;4 

Miscellaneous  Examples                               -            ■                        -  GC 

American  Money               ---...  c,1 

Definitions      -            -                                    •            -            -            -  07 

Table        ■  •            -            •■        -            -            -            .            .  q-j 

Examples  in  Addition            ■■-...  c,<} 

Subtraction             -            -            ...  70 

Multiplication  -  ■ 

Division      .  .  .  .  ^  . 

Miscellaneous  Examples       --•... 


CONTENTS. 


HIGHER    ARITHMETIC. 

Page- 

Defixitioks      -           -           -           -           -           -           -           -  76 

CoMrouxD  Addition           -            -            ,            -            -            -  76 

Compound  Subtraction           -            -            -            -            -            -  80 

Compound  Multiplication            .....  83 

Compound  Division     ...            -            -            -            -  85 

PiiOPKRTiES  OF  Numbers               .....  88 

•Prime  Factors            •.....--  89 

Greatest  Common  Divisor           •            -            -            -            -  90 

Least  Common  Multiple        ...---  91 

Miscellaneous  Examples             -            -            -            -            .  92 


FRACTIONAL     ARITHMETIC. 

Definitions      ....-.--  93 

Vulgar  Fractions             ......  93 

Various  Kinds  and  Examples          -            -            -            -            -  93 

Miscellaneous^  ExAMPLlss             •            -            -            -            -  103 

Decimal  Fractions     ....-••-  105 

Definitions            -            -            •            -            -            -            -  105 

Table               -            -            ■         .   -            -            -            -            -  105 

Explanation  op  Fractions          .----.  106 

Notation          -            -            -            •            -            ■            -            -  106 

Addition                 .......  107 

Subtraction     -            -            -            -            ...            .            -  108 

Multiplication      -            -            -            -            ■            -            -  109 

Division           .....---  110 

Reduction               •            -            -            -            -            -            -  HI 

Miscellaneous  E.xamples       -            -            -            -       .     •            -  114 

Circulating  Decimals       -            -            -            ■            -            •  116 

Continued  Fractions   .         •           -           •           -          .-           -  117 

Duodecimals           •            -            -            -            -            ■            -  118 

Analysis          -            -            -            •            -            -            -            -  522 


CONTENTS.  XI 

PrvOrM^IlTIONAL     ARITHMETIC. 

Page. 
Ratio  Axn  I'soi-ortiov,  oh  Simtlk  Rule  of  Tiiijer  -  -       12 1 

CoJiPouND  Pi;oi'oiiTio\,  oi{  Douiu.B  Rule  ov  Tureu      -  -  •  127 


COMMERCIAL     ARITHMETIC. 

IXTERKST,    SlMI'LI';             -                 -                 -                 -                 •                 -  -         130 

i>fTi:REST,    CoMIMUXl)               -                 '                 -                 -                 -                 -  1?,5 

Discount          -            -            -            -            -            -            -  -      ];i8 

COM.MIISION                    -                 -                 -        ^        -                 -                                  -  l:-?!) 
FELI.OWSliU'    ()!t    PAIlTNK.r.SIIIP,    SlMPLK                  ....          141 

FELLOWSiKI'   t^l    i'aU7N-!;i;SHII',    DoUDLE                          -                  -                  -  I'l?) 

In'suraNi  K                      -             -            -             -            -             -  -141 

Profit  and  Loss                -     ,       -            .         •  .            .            .  i/i", 

Equation  (■'■•  INym  nts          -            •            -            -            -  -       149 

Barter                                 -            -            -            -            -            -  lijl 

Practu-i:  ........       152 

ExCI!AN!.r.                   -              -              -              -         •      -              -              -  134 

Guaci.vg             -            -            -            -         .    -            -            -  -       1.09 

ToxxAui:      -..--.--  161 

Anxuitii>                      -            -            •            -            -            -  -       163  • 

AlLIGATIkN,    >iK,UAL             ......  1G5 

Allioation.  .Xi.tiirxate          -            -            -            -         *   -  -       10(5 

Tark  or  .\t.'    -.v  V  ,-i  K         ......  1G7 

Position,  h;\)     .:         -            -            -            •            -            -  -       1G9 

Position,  Douulk                                        -            -            -            •  170 
MiSCELLAV!  <•' -■   '-    \    ,   .:.s        ......       172- 


)rCAL  ARITHMETIC. 

Intolution                   -            -  -            -            -            -  -       174' 

EvOLUTI.pN,-        :     lOxTRA'.TION    01'  SqUARE    RooT        -                   ■  •                   17t) 

Applicaiidv      1    ^j'^uake  Hoot  -             -             •             •  -       179 

CucR  R.ni;-              -            -  -           -                        •  -            18:J 

E.VTRArTiov  oi"  ('liiiE  Root  -             -             •             -  -       IHt 

MiSCELLA  -                '"     \Mi'LK3  .....  Ib'J 


CONTENTS. 


MISCELLANEOUS    ARITHMETIC. 

Page. 

AuITHMFViOAL  PllOGKESSION        ,             .               .               ...  187 

GEOMSTiaCAL  Progression             -            -            -            -            -  190 

PsKMUTATJOJf   AND    COMBINATION              -            ,    -                 -                 -                 -  191 

MuKtiCRATION                -                  -                  -                  -                  -                  -                  -  193 

MiscKi.i.ANEous  Examples       ------  202 


APPENDIX, 

BOOK-KEEPING  BY   SINGLE  ENTKY. 

ran  l-)Av  Book  -  -  -  -  -  -  207 

Tnio  LEn(5E!i  -  -   .         -  -  -  -  -  207 

TiM^  CU-.i!  Book  ...----  207 

Ti  Book ■         -  207 

.Tu:,  i>ii.!.;-  .\>:n  Notes- Payable         -  .  .  -  -  207 

The  Bills  axi>  Notes  Receivable  -  -  -  207 

Rkmauks  on  Noft;s      -  -  -  -  -     "      -  208 

C():MMKn(jAL  FoKMS  -  -  -  -  -     .       -  .  209 

Ti;;-   '  •>  ::i  of  Day  Book         -         ,   -  -  •  -  -  211 

Til  10  L'i^a.'i  OF  Ledger        ------  213 

TiiE  FoFiM  oi-'-  Cash  Book       ------  215 

Tu!-:  FoKM  OF  Baj[;k  Book  .  .  .  .  -^  216 

iiiE  Fi>KM  oi'^  Bills  and  Notes  Payable     -  -  -  -  217 

Tii.    ''■:ni:i  or  Bills  a.nd  Notes  RECEivAiLt:      -  -  -  218 


COMMON    ARITHMETIC. 


SECTION    I. 

SIMPLE  NUMBERS. 


Article  \,    Arithmedc — a  Greek  derivative — signifies  Antiiiiutii!  d^-- 
simply  the  ajt  of  numbers.     It  h  now  understood  to  com- 
prehend  in   its  exprosision   the   science   or   knowledge    of 
computation. 

2,  A  number  is  what  is  used  to  describe  a  quantity,  and  wimt  m  nmniKr 
is  either  a  unit,  as  the  number  omc  ;  or  a  collection  of  linits, 
as  Uco,  ten,  one  hundred. 

3*  A  number  is  either  simple  or  compound.     It  is  simple  Kumhcrs  sim- 
when  it  expresses  a  single  collection  of  things,  vis  jive  trees  :  |j!^„X   "^"'^ 
compound,  a  collection  of  varieties*,  as  Jive   treeft  and.  six 
apples. 

i.   Arithmetical  science  shows  the  values  and  connections  ^^I'l'^'^'^jJJjj  .j,., 
of  numbers  ;  art,  their  structure,  either  in  complex  form,  or  explained. 
common  relation. 

5<  An  arithmetical  operation  is  icorlrdonehy  i\\Q  employ-  An  oppration: 
nient  of  numbers  ;  the  result  of  the  work  is  the  ansicer. 

6«  A  rule  is  a  direction  to  a  result;  and  a  sum,  the  pro-  .\  rule;  a  sum. 
posed  problem  for  the  exercise  ol'  a  rule. 

7,  The  analysis  6?  a  sum  is  its  separation  info  component  \  ■*"'"  ""'*'■ 

^  '  Vzod. 

parts.. 

8.  There  are  six  diiferent  names  in  arithmetical  u^e,  and  tI"'  manber  of 

,  ,  .     '"  ,  .  .  1   •    ,        77  the  arithmotiC'il 

what  they  express  is  more  or  less  introauced  into  alt  opera-  b.isis,  and  ex- 
t ions  not  si mphi  elementary :  these  are,.  '*^"'^' 

9,  JS'otat ion,' Numeration,  Addition,  Subtraction,  Mvltipli-  ;Y,!!!'^me's*'-^"" 
ration  and  Division. 

10.  Notation — the  simplest  expression  of  a  number —  Notation  d«- 
sfeows  how  to  read  or  write  an  arithmetical  proposition. 

lit  The  forn»  in  which  an  arithmetical    proposition    is  Tinee  forms  oi 
given,  is  in  letters,  figures,  ox  tcords :  the  first  is  known  as  notation. 
Roman  notation  ;  the  second,  Arabic  ;  the  third,  Verbal,  or, 
as  when  we  \\\\t<?  five,  Jifty,  five  hundred ^ 

12.   The  Roman  Notation  employs  seven  capital  letters  :  iipmau  nota- 
thus  I,  expresses  one;  V,  five  ;  X,  ten  ;  L,  fifty;  C,  ojie^""- 
hundred  ;  D,  five  hundred  ;  and  M .  nnr  thousand.     From 
these  we  have  the  following 
1 


NOTATION. 


13.        TABLE  OF  ROMAN  NUMERALS. 


1. 

1. 

XXIV. 

24, 

Boiiian  mi- 

mpvals. 

IL 

2. 

XXV. 

25. 

in. 

3. 

XXIX. 

29. 

IV. 

4. 

XXX. 

30. 

y. 

5. 

.  XL. 

40. 

VI. 

6. 

L. 

50. 

VII. 

7. 

LXX. 

70. 

VIII. 

8. 

XC. 

•    90. 

IX. 

9. 

C. 

100. 

X. 

10. 

CCC. 

300. 

XI. 

11. 

D. 

500. 

XII. 

12. 

DCCCC. 

900. 

XIII. 

13. 

M. 

1000. 

XIV. 

14. 

MD. 

1500. 

XV. 

15. 

MDC. 

1606. 

XVL 

16. 

MDCLXV. 

1665. 

XVII. 

17. 

MDCCXLIX. 

1749. 

XVIII; 

18. 

MDCCCXVI. 

1816. 

XIX. 

19. 

MDCCCLVn. 

1857. 

XX. 

2m 

MDCCCLXn. 

1862. 

XXII. 

22. 

MM.            ,    : 

:  2000. 

Cemu  express      14.   When  two 'or  move  equivalent  letters  are  united,  ov 

Monsof  RoiiiMii  when  one  of  less  value  follow.s  one  of  greatej,  we  have  an 

(expression  of  their  connected  '.vorth  ;  thus,  XX  equals  20  ; 

XLV,  45  •  but  when  a  letter  not  e<juivalent  comes  bpr.ween 

two  of  oreater  value,  it  is  to  he  suhLracled,  or  taken  from 

that  which  follows;  while  its  remainder  must  be  added  to 

that  which  preceded  it ;  thus,  XIV  equals  14  ;  CXL,  140. 

(viuuu  c-xpi-fs-      15.  Sometimes  a  letter  of  lower  value  stands  at  the  left 

t-!nus(^xplaine<).  of  one  of   higher;  then  the  difference  of  their  worth  is 

expressed  ;  thus  IX,  or  one  from  ten,  equals  9  ;  XL,  or  ten 

from  fifty  equals  40. 


I'jxeii'ises  in  Ro- 
jir:m  notation. 


10.  EXERCISES    IN    ROMAN    NOXATION. 

Six, 

Eight, 

Eighteen, 

Twenty. eight, 

Thirty^ 

Forty-nine, 

Eighty -four. 

One  hundred  and  ten. 

Five  hundred  and  thirty-three, 


VI. 


XLfX. 


NOTATIOiV. 


.'i 


"  Six  hundred  and  nine,  DCIX. 

Seven  hundred  and  fourteen, 

One  thousand  and  nineteen,  •         . 

One  thousand  eight  hundred  and  sixty-two. 

17.  Arabic  Notation  is  the  method  to  express  numbers  ^i^''"'-  noiitti-in. 
hy Jigurex.     Ten  are  used,  but  the  first,  0,  is  called  iidis- 
criminatelj  a  zero,  cipher,  or  navght.     When  by  itself  it  The  tiphoi. 
signifies  nothing  ;  but  when  comhined  loith  other  figures,  it  nnd"ith\;[il!!;. 
is  in  name  alone  a  cipher.     The  figures,  exclusive  of  the  0,  , 

are, 

18.  1,        2i         3.  4,         5,         6,         7,  8,         9,  Figures   ami 

one,  two,  three,  four,  five,    six,  seven,  eight,  nine,    1,^10^""^- 
and  ench  represents  the  number  written  beneath,  as  just  "'''■*' 
given. 

19.  There  is  no  single  figure  to  express  the  number  leU.  tuo  manner  ot 
Hence  we  are  obliged  to  place  an  0  at  the  right  hand  of  1  ;  fig"res^  *""  '" 
thus,  \Q  is  ten. 

20.  The   value    of  a   figure    is  'increased  ten-fold   by  How  values *.iw 
removing  it  one  place  towards  the  left;  a  hundredfold  by  ""■'•'^a=f'' 
removing  it  two  places  ;  a.  thousand,  by  removing  it  three, 

and  so  on.  ^ 

21.  The  figures  of  large   numbers  are  more  easily  read  separation  <.r 
when  separated  by  commas  into  periods  of  three.  numbert--. 

22.  Bythe  method  in  common  use*  the  first,  or  right-hand  Naniesaudcou- 
jpc7*io(?,  contains  7/m7s,  tens  and.  hundreds,  ^nA  is  known  as-ifraUng  pe?imls- 
the  period  of  unity;  the  seconci  contains  thousaiids,  tens  of 
f.kousands,  and  hundreds  of  thousa7ids,  and  is  known  as  the 

period  of  thousands  ;  and  the  third  contains  millions,  tens  of 
millions,  and  hundreds  of  millions ;  and  so  on,  according  to 
the  following 


23. 


NUMERATION  TABLE. 


*« 

m 

s 

S 

o 

o 

3  T3 

w 

"*^           r> 

— .     CO 

o  e 

S 

.2   ' 

^.1 

f    3 

z5 

t—  zs 

«.-  rs 

^  2 

• 

'i^ 

■o  ~ 

O  "- 

o  -°  ,-r 

•M' 

tc  W 

«^. 

tB" 

«  o 

to 

C 

—  ^  «i 

">  — 

"O    tn 

_o 

-O    tn 

c 

^    *    3 

TS    ot'   trv 

c   = 

5  = 

<— » 

=    C    o 

s=  •- 

S  'Z 

.E  <^ 

•^ 

3    <B 

■  ~« 

3    (U  .X3 

3    «    C 

Hh 

K^ 

CQ 

,  Kr^ 

s 

KHE^  ■ 

ffiHt) 

5  9, 

6  9 

5, 

5  6 

3, 

8  0  1, 

0  1  0 

5th  period,  4th  period,    3d  period, 
Trillions.      Billions.       Millions'. 


2d  period,    1st  period, 
Thousands.       Units. 


The  numeration 
tnhlo 


Vuiueoi  thegiv-      24.  Expressed  in  words,  the  value  of  the  figures  in  iRp 
entaWe.  above  table,' is  fifty-uiui*  trillions,  six  hundred  and  ninct>y-five 

billions,  five  hundred  and  sixty-three  millions,  eight  hiwidred 

and  one  thousand,  and  ten. 
The  table  ex-         25.  By  the  adoption  of  a  name  for  further  periods,  this 
^1"^®*^'^"^*'^^' table  can  be  indefinitely  extended.     For  common  purposes, 

■  it  is  sufficiently  large. 
Numeration  (|.e-      26.  Numeration   is   simply  the  art  of  reading  numbers 

that  have  been  expressedby  figures. 
i<'igiues.ot  o:i(  h      27.  .Each  period  always  contjiins  three  figures,  except  the 
ponort.  j^^^^  which  may  have  onefigure,  or  two  or  three  figures.     In 

the  table  given,  it  will  be  seen,  that  in  the  fifth  period,  no 

hundreds  are  given. 
■I'o  I'-ii-i  iinni-         28.  To  read  numbers  :  • 

I.  Separate,  by  a  comma,  the  number  into  periods  of  three 

figures  each,  beginning  at  the  right  hand. 

If.  Name  the  order  of  each  figure,  beginning  at  the  right 

hand  ;  as  units,   tens,  hundreds,  and  so  on,  to  the   extent. 

required. 

III.  Then  commence  at  the  left  hand,  and  read  each  period 

as  if  it  stood  alone. 

EXERCISES    IN    A'UMEKATION. 

29*  Let  the  pupil  sejjarafe  by  commas  the  follovCing  num- 
Ifers,  viz  "^ 

468765201 

32575654321 

718950 

985674  M2 

6785403 

25fi87J45 

112543764 

•32789532140 

.3642005428 

5001400 

30.  To  write  ntiinbers  : 

I.  Commeuce  at  the*  'f'ft.  hand,  and  write  each  ptriod  in 
ordfr. 

II.  When  a  fiill  vticant  ptMJod  occurs,  ciphers  must  occupy 
the  spasce. 

-, . ■ 1 : : „  , 

Note. — With-  a  little  care,  the  pupil  will  be  able  to  ■write  any 
number.  Let  him  remember  that  each  period  is  to  be  expressed. 
When  no  nuiTiber  i's  given,  a  cipher  must  be  emplovod.  No  unit,  ten. 
or  himdred  must  be  left  Avithour  expression. 


I'.xeri-i.sos  in 

1. 

iinmevation. 

2. 

3. 

4. 

5. 

6. 

7. 

■   8. 

9. 

10. 

1 .1  wi'ifp  ninn- 

m 

48-   i 

11. 

•127 

12. 

.5«54    • 

13. 

9705 

14. 

32693 

15. 

57865 

16. 

69432     ! 

17. 

234356 

18. 

78.5643     1 

19. 

305400     ! 

20. 

>U?IERATION. 


EXA?1PLES. 


31. 

1. 

■3. 

3. 

4. 

5. 

6. 

7. 

3. 

9. 
10. 
11. 
12. 

13. 


14. 
15. 

16. 

17. 

18. 

19. 


20. 


30 


602 


100,000 


Write  the  following  in  figures: 

Twenty-five, 

Thirty, 

Seventy-eight, 

Four  hundred, 

Six  hundred  and  two, 

Nine  hundred  and  sixty-five, 

One  thousand  and. one, 

Fifty-three  hundred  and  fifty-six, 

One  hundred  thousand, 

One  hundred  thousand  and  eishty,' 

Six  hundred  tliousand  and  two. 

One  million,  three  hundred  and  twenty-one  thou- 
sand. 

Two  hundred  and  twenty. five  billions,  four  bun- 
dred  and  sixty-three  millions,  seven  hundred 
and  ninety-eight  thousand,  two  hundred  and 
thirty-four. 

Three  billions  and  si\ty-five,  3,000,000,065 

Nine  billions,  two  millions,  twenty-five  thousand, 
one  hundred, 

Forty  billions,  one  hundred  and  twenty-seven  mill- 
ions, and  ninety. nine. 

Three  trillions,  sixty-one  thousand,  and  seven. 

One  hundred  and  twenty  billions,  one  thousand, 
and  one  hundred. 

Two  "billions,  twenty,  two  millions,  two  hundred  and 
twenty-two  thousand,  two  hundred  and  twenty- 
two,  2,022,222,222 

One  trillion,  one  hundred  thousand. 


l".x;imples  lor 
practice  in  tim- 
iDcration. 


NOTATION  AND   NUMERATION, 


1.  The  earliest   method  of  denoting  numli^rs  was  prob-  j,,,,  ,;,,||, ,( 
ably  that  of  renresonling  each  unit  by  a  separate  siiin.     No  ni.tiio.i  of .». 

•,.     ,,  '     .       .        °.     J  /        1  .•!  ci,         .       I-     noting  Tiuni- 

perlfctly  convenient   method   was   found,  until  ihc  Arabic  i)«>ri-. 
figures,  or  Digits,  and  the  present   Decimal   system  were 
employed.     These  figures,  0, 1,  2,  .3,  4,  5,  6,  7,  8,  9,^denote  wimi  tho  un- 
nothing,  one  unit,. two  units,  three  units,  and  so  on.  lloto."  " '^' 

2.  To  express  numbers  in  excess  of  9,  availment  is  made  to  exj>res.«  » 
'if  the  law  that  assigns  higher  values  to  figures,  according  ^,"'^""  '"  '"*"  " 
lo  their  position.     According  to  (his  law,  any  figure  at  the 


KOTATION    AND    NUMERATION. 


A  law  in  fijrnrei- 


The  result  of 
the  position. 


Different  value: 
expressed  by 
the  figure  1. 


The  zero— its 
nse. 


Uaity  of  tlie 
second  order. 

Third  order. 


lnterniedi;Ui' 
numbers. 


A  more  consis- 
tent nomencla- 
ture. , 


The  numeni- 
tion  table. 


left  of  another  figure  expresses  ten  times  the  value  that  it 
would  express  if  it  ©ccupied  the  place  of  the  figure  at  it.« 
right.  Consequently,  a  higher  srder  of  units  arises  in  suc- 
cession. 

3.  In  illustration  of  this  law,  let  the  different  values 
.expressed  by  the  figure  1  be  noticed.  Standing  by  itself,  or 
at  the  right  hand  of  other  figures,  1  represents  1  unit  of  the 
first  order  ;  in  the  second  place  towards  the  left,  thus  10,  it 
represents  1  ten,  which  is  one  unit  of  the  second  degree  ; 
when  in  the  third  place,  thus,  100,  it  represents  1  hundred; 
which  is  one  unit  of  the  third  order,  and  so  on.  The  ciphers 
here  employed  are.  without  value  in  themselves.  They  are 
only  used  to  occupy  a  place. 

4.  The  units  of  the  second  order,  or  the  tens,  are  suc- 
cessively named  10,  ten,  20,  twenty,  30,  thirty,  and  so  onto 
90,  ninety.  The  units  of  the  third  order,  or  tlie  hundreds, 
are  named  100,  one  hundred,  200,  two  hundred,  300,  three 
hundred,  and  so 'on  to  900,  nine  hundred.  The  numbers 
between  10  apd  20  are  named  11,  eleven,  12,  twelve,  13, 
thirteen,  and  so  on  to  19,  nineteen.  The  intermediate  num- 
bers in  other  tens  are  similarly  denoted,  but  their  designa- 
tion is  taken  from  the  names  of  their  respective  units ;  thus. 
21,' twenty-one,  22,  twenty-two,  23,  twenty -three,  and  so  on  : 
31,  thirty-one,  32,  thirty-two,  &c.;  41,  forty-one,  &c.  Com- 
pound names  applied  to  the  number.s  between  10  and  20, 
might  be  more  consistent,  thus,  11,  ten-one,  12,  ten-two,  13, 
ten-three,  &c.;  but  those  in  common  use  are  sufficiently 
intelligible.  .   • 

5.  As  the  first  three  places  of  figures  are  appropriated  to 
simple  units,  tens  and  hundreds,  so  every  succeeding  three 
places  are  appropriated  to  the  units,  tens  and  hundreds  of" 
higher  denominations.  Adopting  a  name  for  every  three 
degree  of  units,  the  table  can  be  indefinitely  extended. 


The  tabk  ex- 
terided. 


0:2:0 


o      .2 


Ul        XJl 


a   -a    b^    »    ^ 


c 

0i 


s     S     S     -c     'S 


233,  104,  395,  473,  258,  333,  573,  820,  765,  569,  321,  560. 


Write  in  figures 

1.  Three  octillions,  four  quintillions,  tvi'o  millions  and 
twenty-nine. 

>2.  Twenty-five  septillions,  three  hundred  and  thirty-five 
quadrillions,  thirty-seven  billions,  two  thousand  and  five. 


ADDITION. 


3.  Ten  decillions,  eight  sextillions,  one  thousand  and  one 
hundred. 

4.  Three  hundred  quadrillions,  four  billions  and  sixty-five. 

5.  Sixty-five  octillions,  thirty-five  trillions  and  ninety-five. 

6.  Four  sextillions,  two  thousand  and  three. 


ADDITION. 

32t   Addition  is  the  putting  together  of  one  or  more  num-  Addition  dc- 
herfi,  to  find  their  amount.  ^^  ' 

33«  The  sign  +  is  called  phis,  and  signifies  more.    When  t'^-  ^>g"  v^^»- 
placed  between  two  numbers,  it  shows  that  addition  is  to  be 
performed  ;  thus,  3-f-4,  is  7. 

34«  The  sign  of  equality  is  =,  and  it  signifies  that  the  The  sign  ..r 
quantities  between  which  it  stands  are  equal  to  each  other  ;  ''i""^''*-^  ^ 
thus,  3+4=7. 

35<  These  signs  are  not  employed  when  the  figures  to  be  wii.n  these 
added  are  placed  vertically  :  thus,    5  ;;'«!;(  ''"•*'  "'« 

4       . 
7 

16 

36*  The  latter  is  the  common  form,  and  when  the  num-  wheutiievi-ni- 
hers  are  more  than  simple  collections  of  units  or  ones,  as  in  '"'  ♦'^'■nnsbo-t. 
the  example,  it  is,  if  not  quite  necessary,  the  most  easy  and 
convenient  arrangement. 

37«  Write  the  numbejs  to  be  added  together  in  order,  t„  .„m  mnn- 
placing  units  under  units,  tens  under  tens,  hundreds  under  '"'' 
Hundreds,  and  so  on.  Draw*  a  line  beneath,  as  in  the  exam- 
ple, and  if  the  amount  or  sum  be  less  than  ten,  set  it  under 
that  column  ;  if  it  be  ten  or  more,  put  down  the  units  or 
ones  as  before,  and  add  the  tens  to  the  next  column.  The 
sum  of  the  last  column  is  to  be  written  in  full. 

38«  Example  1. — Add  together  the  numbers  50,  46,  68. 

When  the  numbers  have  been  placed  as 
directed,  the  right  hand  or  unit  column  is 
first  added  ;  thus,  8  and  6  and  0,  are  14 
uiiits,  or  I  ten  and  4  units.  The  4  is  then 
]lut  tmdor  the  first  or  unit  column,  and  the 
1  ten  is  added  to  the  second  column,  or  the 
column  of  tens  ;  thus,  1  and  6  are  7,  and  4  arc  11,  and  5 
are  16  tpn=,  or  1  hundred  and  6  tons.     The  6  tens  is  placed 


OPERATION. 
50 

Rxplauation  of 
tho  way  to  ndd 

46 

68 

164 


ADDITIOJS'. 


under  the  second  column,  or  the  column  of  tens,  and  the  I 
hundred,  which,  if  there  had  been  another  column,  would 
have  been  added  to  that,  is  set  at  the  left  hand  of  the  6. 


Sums  for.prar- 
tjcoip  addition. 

(2.) 

58 

416 

(3.) 

5 

421 

(4.) 
5936 
1862 

(5.) 
3650 
2905 

233 

3856 

21 

3217 

Ans.  707  Am 

9.  4282 

Ans,  7819 

Ans. 

11 

1  1 

1  1 

* 

(6.). 
575 

(7.) 
3360 

(8.) 
56780 

(9.) 
1309652 

8026 

4527 

2300 

275943 

32 

9802 

45695 

322568 

232 

3765 

37821 

534576 

7495 

4930 

26308 

323355 

0 

Ans. 

2769 

39376 
-» 

486789 

.(10.) 
3567820 

(11.) 
5 

(12.) 
39605067 

495323 

74 

4214503 

36532 

532 

7809 

4789 

4789 

890 

532 

36532 

20 

74 

495323 

5060 

•    5 

' 

3567820 

» 

25 

Ans, 

PROOF. 

A  shorter  * 
of  addine. 


Reading 


Uu' 


39.  To  find  if  the  columns  have  been  added  correctly, 
the  simplest  method  is  to  observe  the  s'ame  process,  only 
commencing  at  the  top,  and  adding  downward.  Ifthe  figures 
correspond,  it  is  the  f^roof  sought  of  correct  addition  ;  if  they 
do  not,  the  variation  shows  evidence  of  error. 

40.  In  adding  a  column,  omit  the  names  of  the  figures.. 
Thus,  in  the  12th  sum,  instead  of  saying  5  and  9  are  14  and 
3  are  17  and  7  are  24,  say  briefly,  5,  14,  17,  24,  and  then 
setting  down  the  4  units*  or  ones,  and  carrying  the  2,  say  4, 
10,  12,  21,  27,  and  so  pn.  This  is  called  reading  the  col- 
umns, and  is  done  by  good  accountants.  Frequent  practicft 
will  render  it  easy. 


*  From  the  Latin  unus,  which  means  one. 


ADDITION.  -  '.} 

Example  (13.)  Add  7959,  6543,  and  3487  together.        Examples  f.r 

(14.)  Add  6954,  78421,5678,  34,5659,  432178,  598765,  i>r»<f'-'e.nai.i. 
>21234,  56789015^and  325.to^ether. 

(15.)  Add  32,  361,  4500,51189,  67891,  3606,48572, 
72684,  30605.4336,  211667, 5870,  59865,  and  999  together. 

(,16.)  Add  71 19, 59435,  625,  32,59601,  43C656,  5987451, 
and  23205  together. 

(17.)  A<id  59590,  607805,  43021,  678015,  3676789, 
39:)43201,  4856729,  and  66  togother. 

(18.)   Add   2005960,  5067520,  495610,   450.  3798510, 

69450,  798504,  3296503,  576531,  723456,  and  777  together. 

.(19.)    Add     325678,    4353791,     56789012,    5394820, 

506070801,  59694534,  3787956,  3950034,  and  25  together. 

(20.)  Add  253651,  8135430,  376932,6554322,  78985607, 
;^59.  6432.  5765432,  and  8  together. 

(21.)  Add  five,  fifty.-five,  one  hundred  and  ton,  one  thou- 
sand and  ten,  two  hundred  and  seveoty-three,  thirty-three 
n^illion^!,  two  thousand  and  ninety-nine,  five  thousand,  and 
three  hundred,  atid  (briy-eight  together.  * 

(22.)  Add  three  trillions,  two  hundred  and  seven  billions, 
five  hundred  and  seventy-five  millions,  si.x  hundred  and  thirty- 
six  thousand,  and  s^ven  hundred  and  eighty-four  together. 

(23.)  Add  three  hundred  and  five  hillinns,  two  hundred 
and  foity-six  millions,  seven  hundred  and  fifty-nine  thousand 
and  eighty  together. 

•   Re>jark. — The  sign  that  denotes  a  dollar,  or  dollars  is  $  ;  Ttiodoii.ir.<.^ii:i. 
thus,  $1  represents  one  dollar  ;  $5,  five  dollars;  $100,  one 
hundred  dollars.  ^  , 

PRACTICAL    EXAMPLES. 

E.VAMPLE  (1.^  There  are  due  to  my  factors  in  Charles-  pnK-tioaJ  vx- 
ton.  fur   money  advanced   on   cotton,  S 150,  to  uiy  grocers, '""'''''■" 
$275,  to  the  machinist  forn  steam-engine,  S050,  to  the  dry- 
goods  merchants,  $159,  to  the  hardware  dealers,  $87,  and 
to  the  Rank  of  the  Slate  of  Sontfi  Carolina,  for  a  note  dis- 
counted, S2895:  what  is  the  amount  of  this  indebtedness? 

(2.)  The  State  of  A' ir<£inia  harl,  in  the  year  1790.  a  pop-  Papulation  of 
ulation  of  748,308;  of  Maryland.  319 '28  ;  of  North  Car-  f,ru!r"^'''" 
lina,  393.751 ;  -of  Tennessee^  .35,791 ;  of  Kentucky.  73,077  ; 
of  Georgia,  82,548  ;  and  of  South  Carolina,  249,073  :  what 
was  the  aggregate  ? 

(3i)  The  same  States  had,  in  18-30,  the  first  named  a 
population  of  1,211.405;  ih"  second,  447,040  ;  the  third, 
737.987;  the  fourth,  681,901;  the  filth,  687,917  ;,ihe 
sixlh.  516  823;  and  the  last.  5^L1''5  :   what  was  the  whole, 


the  Northern 

States. 


10  ■    ADDITION. 

Free  jjopuia-        .(^O  The  States  forming  the  Southern  Confeder»cy,  iu 
^'Ste  st^aTefin'  1861,  had,  according  to  the  last  census— South  Carolina,  a 
hhl.         "       free    population  of  301,271 ;  Georgia,*  595,079  ;   Florida. 
78,686  ;   Alabama,   529,164  ;    Mississippi^  354,699  ;  Lou- 
isiana,   376,913 ;    Arkansas,   324,233;    Texas,   420,651; 
Virginia,  1,105,196:  what  was  the  total  ?  ^i 

Siave  wvpuia-  (5.)  The  same  Confederate  States  had.  at  that  date,  a 
iTerite'state""  ^'^^'^  population— South  Carolina,  of  402.541  ;  Georgia, 
,u'i86L  '  "  460,232  ;  Florida,  61,753  ;  Alabama,  435.132  ;  Mississippi, 
4.36,696;  Louisiana,  332,520  ;  Arkansas.lir,104 ;  Texas. 
180.388  ;  Virginia,  490,887 :  what  was  the  entire  number  I 
The  United  (6.)  The   United   States  has  9,334  miles  of  coast ;  the 

Idlrite'states'"  Confederate  States,  28,803 :  what  is  the  whole  ? 
sea-coast.  (7.)  The  State  of  Maine  has  of  sea-coast  and  shores,  of 

'rhej5ea-coast  of  bays,   sounds,  dsc,  and  of  rivers  to  tide-head,  a  total  of 
miles,  <  '  2,452 

The  State  of"Ne<V  Hampshire,  74 

"        ,  "      Massachusetts,  1,906 

■      "  '*      Rhode  Island,  440 

"  "      Connecticut,  1,327 

«      New  York,  ■  2,057 

"  "     New  Jersey,  571 

'•  "      Pennsylvania,  106 

What  is  the  whole  number? 

seu-coast  of  (8.)  The  Slate  of  Delaware  has  of  sea-coast  and  "shores, 

valnlj^ivan'^a?'  ^^  ^^ys,  sounds,  «fec.,  and  of  rivers  to  head  of  tide,  a  total 

of  miles,  *                            671 

The  State  of  Maryland,  4,453 

"      Virginia,  2.372 

"           "      North.  Carolina.  2,780 

"           "      South  Carolina,  1,256 

"           "      Georgia,  924 

"      Florida,.  4,885 

"      Alabama,  630 

*'           "      Mississippi,'  385 

"           "      Louisiana,  3.147 

"           "      Texas,  .             3.069 
What  is  the  total  ? 

(9.)  The  exports  of  the  United  States  during  the  fiscal 
year  of  1859,  were,  of  dopiestic  produce,  chiefly  from  the 
sl^ve  States — 


APDITIOX.  II 

Cotton,                  .                      ■  $161,434,923       Exports;  oi  u.. 

Rice  and  other  vegetaWe  productions,  24.046,752      ^'".'ot.'l  •^'•'*'"' 

lobacco,  21.074,038 

Manufactures,  33.853,660 

Animal  products,  15,549,817 
•  Products  of  the  forest.                              ~        14,489,-^00 

"           '•     •  sea,  4.462,974 
What  was  the  amount  ? 

(10.)  The  imports  of  the  United  States,  the  same  year,  i^portsof  cr.  - 

were,  Dutiable  £;oods.  ^259,047,014      t.^a  states.  T'si'. 

Free  goods,  72,286,327 

Specie  and  b.i!lion,  7,434,789  ' 

Domestic  produce,                               '  278,392,080 

Koreiijn  produce  at;d  merchandise,  14,509,971  . 

Domestic  specie  and  biiHio'n,  57.502,305 

Foreiiin         "                "  6,885,106 
What'  was  the  amotmt  ? 

(11.)  The  imports 'into  the  United  .States  during  1859  items  of  in,-  ' 

were —  port?,  isoo. 

Woo!  and  woolens,  S37.966,910 

Cottons.  *  29,830,364 

Silks,'                                   ■  29,487,513 

Flax  and  linens,  10,487.891 

Tea,  7,388,741 

Coffee,  25,085,696               » 
Rawhides.                                             .       13.011,326 

Tin  plates.,  5,331,147 

Molasses.                             '  5.062.850 

Sugars,  brown,                      .          •  .30,471,302 

Tobacco  and  segar^.'                   .  6,267,855 

fron,  9.088,239 

Steel,  3.09M35. 

Brandy,  wine,  kc,  8.919.885 
What  was  the,araount  ? 

(12.)  Accordins  to  the  late  tables,  the  pof)ulation  of  the  Popuiatir.u  rj 

-^lobo  in  1859,  wa".s —  the  ^'loVe. 

?>urope.  272.000,000 

A^sia,  755;000,000 

America.  20%000,000 

Africa,  59,000,000 

Australia,  (kc,  2,000,000 
What  was  the  entire  number  : 


12  .  ADDITION. 


Population  of        (13-)  According  to  races,  the  population  of  the  globe  was, 

the  ranes.  j^  I860-- 

Caucasian,  369,000,000 

Mongolian,  522,000,000 

Ethiopian,  196,000,000 

American,  1,000,000 

Malay,  ^00,000.000 

What  was  the  total? 
Population  reii-      (14.)  In  religion,  the  population  of  the'  globe  is  thus 

Kiously  divided.  (]jyj(jgj^  •  . 

Christians— Protestants,  89,000.000 

Romish  Church,  .  170,000,000 

Jews,  5,000,000 

Mohammedans,  '            160,000.000 

Heathen,  788,000,000 

What  is  the  sum  ? 

Lo.sse.s  of  the  (^^O  ^"  ^^^  '^^^^  ^^  ^^^  Revolution,  the  losses  sustained 
British  and  by  the  English  and  Americans,  including  in  some  cases  the 
the  Revolution-  wounded,  and  the  surrenders  at  Saratoga  and  Yorktown, 
iry  Wnr.  hsLve  been  computed  as  follows  : 

Lexington,  April  19, 1775, 
Bunker  Hill,  June  17,  1775, 
<•  Fort  Mouhrie,  June  28,  1776, 

Flatbush,  Augiist  12,  1776, 
White  Plains,  August  26,  1776, 
Trenton,  December  25,  1776, 
Princeton,  January's,  1777, 
Hubbardstown,  Aug.  16  and  1 7, 1777, 
Bennington,  August,  16,  1777, 
Brandy  wine,  September  11,  1777, 
Stillwater,  September  17,  .1777, 
Germantown,  October  4,  1777,     .  ^ 
•    Saratoga,  Oct.  17, 1777,  surrendered, 
Red  Hook,  October  22,  1777, 
Monmouth,  June  25,  1778, 
Rhode  Island,  August  27,  1778, 
Briar  Creek,  March  30, 1779, 
Stoney  Point,  July  15,  1779, 
Camden,  August  16,  1779, 
King's  Mountain,  October  1,  1780, 


British. 

American. 

Loss. 

Loss. 

273 

-93 

1054 

454 

205 

11 

400 

200 

409 

400 

1000 

9 

400 

100 

800  ■ 

800 

800 

100 

500 

1100 

600 

350 

600 

1200 

6752 

500 

32 

400 

130 

260 

211 

13 

400 

600 

100 

375 

610 

950 

96 

.     ADDITION.  •        1'^ 

Cowpen^,  January  17,  1781,                     800  72 

Guilford  Court  Houso.  MarrI,  15, 1781,     532.  ,  -IPO  ' 

Hobkirk  Hill,  April  25,  1781.     '              400  400 

Eutaw  Springs,  September,  1781,           1000  550 

Yorktown,  October,  1781,  surreiKlered,  7072         

What  was  the  entire  loss  on  either  side? 

(16.)  Gen.  Washington  was  born  A.  D.  1732,  and  lived 
67  years  :  in  what  year  did  he  die  ? 

(17.)  How  many  days  are  there  in  the  twelve  calendar 
months,  January  having  31,  February  2S.  March  31,  April 
30.  May  31,  June  30,  July  31,  August  31,  September  30, 
October  31,  November  30,"  December  31  ? 

(18.)  William  the  Conqueror  began  to  reifrn  in  England 
in  the  year  1006,  and  rrigned  21  years;  William  II, ,  13 
years  ;  Heurvl..  15  ;  Stephen,  39;  Henry  11.,  35;  Richard  I., 
iO  ;  John,  17  ;  Henry  HI.,  56  ;  Edward  I.,  35  ;  Edward  11., 
20  ;  Edward  HI.,  50  ;  Richard  H.,  2'3i  In  what  year  was 
Richard  dethroned  ? 

(19.)  From  the  creation  of  tlie  world  to  the  flood  wa.^ 
1656  years  ;  front?  that  time  to  the  building  of  SolomonV 
Temple,  1336  years;  thence  to  the  birth  of  our  Saviour. 
1003  years  :  in  what  year  of  the  world  was  our' Lord  born"? 

(20.)  If  you  invested  in  the  Hank  of  Georgia,  $5000  ;  in 
the  Mississippi  State  Funds,  $2550  ;  in  the  Savannah  City 
Stock,  $1550  ;  in  the  Southwestern  Railroad  Company, 
82000 ;  and  in  the  Port  Royal  Railroad,  85500  :  what 
would  be  the  amount  thus  placed  ? 

(21.)  A  father  in  his  will  leaves  to  his  eldest  daughter, 
^5000 ;  to  his  youngest,  $3550  ;  to  his  four  sons,  $300 
each  ;  and  to  them  also,  severally,  the  same  sum  left  to  his 
eldest  daughter  :   what  was  the  entire  amount  ?    • 

(22.)  A.  planter  sent  to  his  factor,  at  one  time,  4  billes  of 
cotton,  which  weighed  1437  pounds;  at  anothor  6,  which 
weighed  2100  ;  at  a  later  date  10,  which  weighed  3575  ; 
and  afterwards  25,  which  weigliod  8000  :  what  was  the 
whole  nu  nber  of  pounds,  and  how  many  bales  ? 

(23.)  A  factor  received  on  consignment  300  busheb  of 
rice;  a  few  days  later  2169;  then  5560  ;  afterwards  4175  ; 
and  last,  2358  :  how  many  bushels  in  all  ? 
'.  (24.)  There  are  60  seconds  in  a  minute,  3000  In  an  hour. 
86,400  in  a  day,  604,800  in  a  week,  2,419,200  in  a  month, 
and  31,557,600  in  a  year:  how  many  are  all  these  combined 
together? 

(25.)  A  merchant  in  Mobile  received  frr  m  New  Orleans, 
ten  casks  of  hardware,  weighing  each  19604-875+1000+ 


It     •  ADDITION. 

1268+999+25614-3248+1545+1862+1300:    what   was 
fhe  .weight  altogether  ? 

(26.)  The  sums  on  the  debit  side  of  a  merchant's  books 
for  three  different  periods  were  as  follows  : 


832.68 


15789.51 

$2134.c0 

$789.61 

1643.42 

4357.00 

5.48 

347.31 

429.66 

14.52 

56.25 

3906  25 

23.07 

3.33 

78.16 
1832  43 

741.50 
259.30 

6789.75 
1150.00 

745.S3 
.  5432.10 

212.13 

567.49 
874.32 

56.15 
2.12 

75.15 

83.33 

16.79 

7128.23 

79.20 
8901.31 
4728.53 

56.02 
47.14 

'     2.25 
9.15 

87.42 

773.19 

940.42 

,940.43 

^544.96 

'  59.75 

366.03 

337.16 

42.58 

267.30 
22.81 

569.87 
24.56, 

18.76 

256.00 

842.15 

1530.21 

.55.02 

47.96 

55.02 

268.34 

•      81.15 

586.75 

367.35 

46.53 

142.04 

2269.54  • 

•487.20 

1678.39 


693. 


1509.99 


Ans. 


The  porioii  Rematik. — In    the   last  three  sums,   the  'period  marks 

''u?«hin"the  between  the  second  and  third  columns,  distinguish  the 
lioiiiir.-^ from      dollfw's  and  cents  ;  thus,  the  bottom  figures  in  the  last  sum, 

1509.99,'  are  read  fifteen  hundred   and  nine  dollars,  and 

ninety-nine  cents. 

^ : _, . -r-, ■ :— 

I'olnmns  to  be  Note. — When  the  columns  of  figures  are  very  long,  let  them  be 
^.iepar.tted  wher.  separated  by  Imes,  as  in  the  last  example ;  and  tlie  separated  amounts 
\pry  long.  placed  by  themselves,  added  for  the  required  answer. 


i     ilH 


ADDITION.  15 

The  census  of  the   United  States,  in  1850,  and  that  of 
1*^60,  were  as  follow?  :  what  were  the  totals  ? 

CENSUS  OF  1850. 
.)    States.  Fr«e.  Slave,  Total. 

Alabama,  428,779    342.844        771,623 

Arkansas,  162.797      47,100        209.897      Kr"*^'" 

California,  92.597  92,597 

Connectic\it,  :nO  792  370,792 

Delaware,  89,242        2,290     .      91,532* 

Florida,  4S.135      39,310  87,445 

Georgia,  524,503    381,682         906,165 

Illinois,  651,470  851,470 

Indiana,  983.416  938,416 

Iowa,  192.214  192.214 

Kansas, 

Kentucky^ 

Louisiana, 

Maine, 

iMaryland, 

Massachusetts, 

Mississippi, 

.'Missouri, 

Michigan, 

Minnesota, 

New  Hampshire, 

Xew  Jersey, 

New  Y6rk, 

North  Carolina, 

Ohio, 

Oregon, 

Pennsylvania, 

Uhode  hlaud,* 

.South  Carolina, 

Tennessee, 

Texas, 

Vermont, 

Virginia, 

VV'isconsin. 

'rKKRITORIES 

Colorado, 
Dakotah, 
Nebraska, 
Xevada, 
Xew  Mexico, 
Ttah, 

Washington, 
Dis.  of  Columbia, 
Ans, 


771.424 

210,981 

^982,405 

272,953 

244,809 

517,762 

583.169 

,583,169 

492.666 

90,368 

583.034 

994,514 

994,514 

296,648 

309,878 

606,526 

594.622 

87,422 

682,044 

397,654 

397,654 

6,077 

6,077 

* 

317.976 

317,976 

489.319 

236 

489,555 

3.097,394 

3,097,394 

5^0,491 

288,549 

869,039 

1,980,329 

1.980,329 

13.294 

13,294 

2.311.786 

2.311,786 

147.545 

147.545 

283.523 

384.984 

668,507 

763.258 

289,459 

1,002,717 

154,431 

58,161 

212,592 

314,120 

314,120 

949,133 

472,528 

1,421,661 

305.,'^91 

305.391 

t'clisus  ivf  t 
U.S.TfiT 
ries  in  is:.' 

61.547 

(51.547 

11,354 

26 

1 1,380 

»    48.020 

8,687 

51.687 

ir 


Census  of  tlie 
U^ite.l  States 


'  I'li.-u?'  Ill",  the 
.r.  S.  'i'crrito- 
•i<'^  111  i»;(i. 


ADDITION. 

■ 

CENSUS  OF  ISOO. 

(i'<:)      i^TWES. 

Free. 

Sl.Wi!. 

TOIAL. 

AlAbrima, 

529,164 

435,132 

964,296 

Arkansas, 

324,323 

111,104 

435,427 

California, 

380,015 

380.015 

Connecticut, 

460,151 

460,151 

Delaware, 

110,420 

1,798 

112,218 

•Florida, 

78,630 

61,753 

•    140.439 

Georgia, 

595,097 

462,230 

1.057.327 

Illinois, 

1,711,753 

1,711,753 

Indiana, 

1,350,479 

1,350,479 

Iowa, 

674,948 

674.948 

Kansas, 

107,110 

107.110 

.  Kentucky, 

930,223 

225,490 

1,1.55,713 

Louisiana, 

376,913 

332,520 

709,433 

Maine,  * 

628  276 

628,276 

Maryland, 

599,846 

87,188 

687,034 

Massachusett 

s,             1,231,065 

1.231.065 

Mississippi, : 

354,699 

436,696 

791,395 

Missouri, 

1,058,352 

111,965 

1,173,317 

Michigiin, 

749,112 

749,112' 

Minnesota, 

102.022 

162,022 

New  Hamps 

hire.           320,072 

326,072 

New  Jersey, 

672.03'I 

672,031 

New  Yojk, 

3,887,542 

3,887,542 

North  Carol i 

na,              061,556 

331,681 

•S82  667 

Ohio, 

2,339,599 

■2,339,599 

Oregon, 

52.464 

52,464 

.  Pennsylvania 

2,906,370 

2,906.370 

Rhode  Island 

L              .  ..  174,621 

174,621 

South  Caroli 

na,              301.271 

402.5M 

.703,812 

Tennessee, 

834083 

.275,784" 

1,109,847 

Texas, 

420,651 

180,388 

601,039 

Vermont, 

315,1 16' 

31.5,110 

Virginia, 

1,105,198 

490,887 

1,596,083 

Wisconsin, 

775.873 

775.873 

Tep.hitories. 

Colorado, 

34,197 

34,197 

Dakotah, 

4.S39 

4.839 

Nebraska, 

■      28,632 

10 

28,842 

Nevada, 

6,857 

G,857 

Now"  Mexico 

03,517 

24 

93,541 

Utah,     . 

40.260 

29 

40.295 

Washington, 

11.578 

11,578 

Dis.o.f"  Colli 

mbia,            71.895 

3.181 

75.07t) 

Aihs.                       . 

RUBTK  ACTION.  17 

The  primary  mode  of  formino;  numbers  by  joinmg  oneThepriucipieoi 
unit  to  another,  and  the  sum  of  those  to  a  third,  and  so  on,  *<i'i'i'o"- 
exhibits  the  principle  of  addition.  When  the  numbers  to 
be  added  consist  of  units  of  several  degrees,  such  as  tens, 
iiundreds,  etc.,  it  is  more  convenient  to  add  together  the 
units  of  each  order  separately  ;  and  since  ten  units  of  any 
Older  mike  one  unit  of  the  next  higher  class,  the  number  of 
tens  in  the  sum  of  each  order  of  units  is  carried  to  the  next 
higher  cla«s  and  added  to  it. 

!f  the  sum  of  the  figures  in  each  column  be  not  in  the  The  renson  for 

excess  of  nine,  the   operation  could   be  equally  well  com- ''5j'l"^""."i? 

II        1  1  !•  •  !•    1  •         /•    I      I  •    .  1  addition  at  the 

mencel  ny  the  addition  ot  the  units  of  the  highest  order  as  right. 

by  that  of  the  simple  units.  But  as  it  is  ofiener  than  other- 
wise the  case,  that  several  of  these  sums  are  in  the  excess 
m  nine,  to  commence  on  the  left  obliges  the  operator  to 
return  back  to  correct  a  figure  already  written,  and  increase 
it  by  as  many  units"  as  shall  be  obtained  from  the  tens  of 
the  following  column.  Hence  it  is  better  in  all  cases  to 
commence  adding  at  the  ri^jht. 


SUBTRACTION. 

41i  Suhtraclion  is  that  operation  which  shows  the  differ'  Subtraction  de- 
ence  between  two  numbers.  ^"*''^' 

42*  'I'he  difference  when  found,  and  added  to  the  smaller  ing^th^  great*" 
number,  gives  the  greater  :  this  also  proves  the  operation,    •jumper;  also, 

43.  'I'he  difference  is  called  the  remainder  ;  th«  greater  j^anjegof  terms 
number,  the  minuend. ;  the  smaller,  the  subtrahend. 

44t  The  sigii  of  subtraction  ( — )  is  called  7ninus,  and  sig-  Thesign minus, 
nifies  less.     When  placed  between  two  numbers,  it  shows 
that  the  right  hand  figure  is  to  be  taken  from  the  one  at  the 
left  ;  thus,  9— 5=4.  thai  is,  nine  ininus  five  equals  fijur :   in  Example  of  its 
this  example  9  is  the  minuend,  .5  ihe  subtrahend,  and  4  the  "^«- 
difference,  or  as  it  is  also  called,  the  remainder. 

45.  To   subtract  a  number,  place  the  less   beneath  the  To  subtract 
greater,  so  that  units   come  under   unit«,  lens  under  tens,  ""'"''•''•^• 
hundreds  under  hundreds,  and  so  on. 

I.  Commence  at  the  right  hand  and  subtract  each  figure 
of  the  lower  line  from  the  one  directly  above,'when  the  one 
above  is  greater  ;  then  set  the  remainder  under  the  horizon- 
tal line  drawn  beneath  the  subtrahend. 

II.  If  the  figure  to  be  subtracted  is  the  greatest,  then 

3 


J-; 


SUBTRACTION. 


add'lO  to  the  figure  in  the  minuend,  and  proceed  as  above 
directed  ;  always  adding- 1  to  the  next  lower  figure,  when 
10  has  been  added  to  the  figure  in  the  previous  minuend. 

Whcnice  the  10      Remaek.— The  10  which  is  added,  when  the  unit  figure 
is  mkeu.  jg  to  bg  increased,  is  1  ten  from  the  next  upper  figure  of  tens  ; 

when  the  tens  figure  is  to   be  increased,  the  ten  is  taken 

from  the  hundreds  figure,  and  so  on. 

This  is  called  borrowing,  and  in  all  cases  when  this  is 

done,  there  must  be  carried  or  paid  to  the  next  subtrahend  1, 

which  is   the  borrowed  number  in  a  different  form,  but  of 

equal  value. 

EXAMPLE  [1.3 

EXPLANATION.  -. 

In  this  example,  we  begin  at  the  units 
column,    and    say  3  from  5  leaves  2. 
Setting  that  down,  we, then,  at  the  sec- 
ond or  tens  column,  say  2  from  6  leaves 
4;  and  having  put  that  down,  we  proceed  with  the  last  or 
hundreds  cfllumn,  and  say  3  from  5  leaves  2,  which,  placed 
by  the  side  of  the  4  completes  the  sum. 


What  is  bor- 


Explanatiou  of         OPERATION. 

ihework.         Minuend,         565 


Subtrahend,     323 
Remainder,     242 


[2] 
Minuend,  954 
Subtrahend,   643 

Remainder, 


659 

538 


W  [5] 

269  545 

137  433 


46.  The  answer  to  example  1  is  found  to  be  right,  by 
adding  the  subtrahend  and  remainder  together.  These 
when  added  give  the  minuend  : 

Subtrahend,    323  ;    . 

Remainder,    242 
Minuend,        565 


EXAMPLE   [6] 
OPERATION.  EXPLANATION. 

Minuend,  834  Here,  we  cannot  take  7  units  from  4 
Subtrahend,  427    units,  but  by  adding  10,  which  makes  the 

minuend  14,  we  can  say  7  from  14  leaves 

Remainder,  407    7.     This  being  put  down,  vvecavry  1  to 

— -    the  next  figure,  and  say  3  from  3  leaves 

Proof,    884    naught.     When  this  has  been  set  down, 

we  proceed  with  the  last  figure,  and  say  4  from  8  leaves  4. 

and  place  it  beside  the  last.     Tlie  sum  is  then  done,  and  i- 

proved  by  the  addition  of  the  subtrahend  and  remainder. 


SUBTRACTION. 


10 


17] 
x\Iin.50G5 
Sub.  487G 

Rem. 

Pr. 


[8]  [9] 

3256         6945 
1337         5938 


[10] 
9654 
8573 


[11]  [12] 

7643        5395 
6374        4236 


EXAMPLE    [13.] 
orERATION.  EXPLANATION. 

.Mimiend,       7650        In  this  example,  with  the  figure  in  the  Explanation. 
Subtrahend,  5761    thousands-plnce,  all  of  the  subtrahend  is 

greater  than  the  minuend.     Of  course 

Remainder,  1889  10  has  to  be  borrowed  and  added  to  each 
figure  of  the  minuend,  except  the  last,  and  1  is  to  be  carried 
or  paid  to  every  figure  in  the  subtrahend,  except  that  with 
which  the  sum  is  commenced.  Thus,  we  say  1  from  10 
leaves  9  ;  then  carrying  1  to  the  6,  we  say  7  from  15  leaves 
8;  carrying  1  to  the  7,  we  say  8  trom  16  leaves  8;  and 
last,  1  to  5  is  6,  and  6  from  7  leaves  1. 


Mill. 
<3ub. 

Rem. 
Pr. 


[19] 

Min.  3653456 
Sub.  2895667 


[14]  [15] 

7S65         8954 

976        6165 


[16]  [17]  [18] 

7532         88542         1045679 
4507         59653  565789 


[20] 
42310105670 

37824306580 


[21] 

1000 
1 


[22] 

60000 
99 


[23] 

45670000 
432506 


Rem. 
Pr. 


24.  From  56754302,  take  4356706. 

25.  From  65374231,  take  5900327. 

26.  From  twenty  thoueand  and  thirty-five,  take  one 
thousand  five  hundred. 

27.  From  one  billion,  two  hundred  and  forty-two  thou- 
sand and  twenty.three,  take  one  million  and  ninety-five, 

28.  376326945-22654178=  ? 

29.  456897003—3955487=  ? 

30.  90005434—785070=  ? 

31.  7000000—3999993=  ? 

32.  58675432—40101011=  ^ 

33.  In  a  certain  example,  the  minuend  is  368  and  the 
subtrahend  209  :  what  is  the  remainder  ? 

34.  Henry  Hudson  sailed  up  the  North  river,  now  called 
after  that  navigator,  the  Hudson,  in  1609  :  how  many  years 
<ince  ? 


20  SUBTRACTION. 

85.  Sjjppose  ^ou  should  borrow  of  a  friend  $2000,  and 
pay  baqk  at  different  times  $1695 :  how  much  is  the 
remaining  indebtedness? 

36.  If  your  income  is  $2500  a  year,  how  much  is  your 
annual  deficiency  if  your  expenditures  amount  to  $2745  ? 

37.  The  battle  of  Fort  Moultrie  was  fought  in  1776 : 
what  number  of  years  have  elapsed  since  that  memorable 
contest  and  the  investment  and  capture  of  Fort  Sumter, 
in  1861? 

38.  The  Revolutionary  war  commenced  in  1775  :  how 
long  is  it  since  that  date  and  the  war  of  1812? 

39.  The  distance  from  the.  earth  to  the  sun  is  called 
95,000,000  miles  ;  the  distance  to  the  moon  240,000  : 
how  many  more  miles  is  it  to  the  sun  than  to. the  moon? 

40.  South  Carolina  passed  the  ordinance  tif  secession, 
December,  1860 :  how  many  years  is  that  era  from  the 
discoTery  of  America  in  1492  ? 

Remark. — Sometimes  it  is  convenient  to  subtract  with- 
out placing  the  subtrahend  beneath  the  minuend.  It  is, 
however,  virtually  so  placed. 


Subtrahend,     957 
Minuend,        2756 

Remainder,     1 799 


41.  Subtract  957  from  2756, 

42.  Subtract  635  from  1650. 

43.  Subtract  342  from  560. 

44.  Subtract  541  from  9650. 

ihe  rationale  Subtraction  is  performed  by  taking  the  units  of  each 
of  subtraction.  (Jegree  in  the  subtrahend  from  those  of  the  corresponding 
degree  in  the  minuend,  and  severally  denoting  the  remain- 
ders.  When  the  units. of  any  degree  in  the  subtrahend 
exceed  those  of  the  same  degree  in  the  minuend,  one  unit 
of  the  next  higher  degree  is  to  be  mentally  joined  to  the 
deficient  place  in  the  minuend,  and  the  units  of  the  higher 
degree  to  be  considered  one  less  than  denotvd.  Another 
method  maybe  adopted  in  this  case:  increa.se  bt)ih  the 
minuend  and  subtrahend  by  the  mental  addition  o(  ten  to 
the  deficient  place  in  the  former,  and  on^  to  tlie  next  hit'her 
degree  6f  units  in  the  latter.  This  method  i>  justifiid  by 
the  selfevident  truth  that  if  two  unequal  quantities  be  equally 
increased,  their  difference  is  the  .•<ame. 
An  example  To  find  the  difference  which  exists  between  two  numbers, 
^d  explana-  vvhen  some  figures  in  the  lower  line  are  greater  than  some 
intheupper,  the. iollowing  process  is  to  be  observed  : 


SUBTRACTION. 


•21 


OPERATION. 

83456 

28784 


Having  arranged  the  numbers,  as  in  the 
example  given,  we  say  4  from  6  leaves  2, 
which  is  written  unrler  th«  units.  Pass- 
ing to  the  cohinin  of  tens,  as  the  lower 
figino  8  is  greuter  than  the  upper  one  5,  it  54672 

cannot  he  siilitracted.  To  overcome  this  difficulty,  we  bor- 
row tnenfally  from  the  hundreds  figure  1  hundred,  which  is 
equal  to  10  ten?,  and  add  to  it  the  5  tens,  which  we  have 
already,  making  it  15  tens;  we  then  say  8  from  15  leaves 
7,  which  is  written  in  the  column  of  tens.  Proceeding  to 
the  column  of  hundreds,  we  observe  that  the  upper  figure 
ought  to  be  diininisht'd  by  1.  as  this  unit  was  borrowed  in 
the  preceding  subtraction  :  we  say,  then,  7  from  3  (or  by 
adding,  whicii  would  he  equivalent,  the  borrowed  1  to  8,  we 
say  8  from  4),  which  is  impossible;  but  we  borrow,  as 
before  1  thousand,  which  equals  10  hundreds,  giving  13 
hundreds,  and  take  7  from  13  which  leaves  6  (or  by  adding 
the  borrowed  unit  to  the  7.  and  taking  8  from  14),  to  be  placed 
in  the  column  of  hundreds.  Passing  to  the  thousands,  8 
cannot  be  taken  from  2  (or,  equivalently,  9  from  3),  b'.'t  8 
from  12  leaves  4  (or  9  from  13),  which  is  written  in  the 
column  of  thousand?.  Lastly,  as  the  figure  8,  of  tens  of 
thousands,  on  account  of  the  1  just  borrowed  ought  to  be 
replaced  by  7,  we  say  2  from  7  leaves  5  (or,  equivalently, 
o  from  8).  'I'hus  the  rerariinder,  or  the  excess  of  the  greater 
number  over  the  less,  is  54672. 

To  understand  how  by  this  method  we  learn  such  result, '^'^'^  rationale. 
it  is  suflicieiit  to  state  that  the  two  numbers  could  be  thus 
ari'ranjred  : 


Tens  of  thousand.-*. 

1st  number.  7 
2d  number,   2 


Thousands. 

Hundreds. 

Tens. 

Unit 

12 

13 

15 

6 

8 

7 

8 

4 

5  4  6  7        2 

It  thus  appears  that  the  upper  number  exceeds  the  lower 
one  l)y  2  units,  7  tens,  6  hundreds,  4  thousands,  and  5  lens 
of  thousands — 6r  exceeds  it  54072  units. 

OPERATION. 

I      lt9    9         Apecondexam- 

300405  '''''• 
158429  ' 


Airain,  to  subtract  the  number  158429  from 
300405. 

As  9  the   units  ficure  of  the  lower  number 
is  larger  than  5,  the  corresponding  figure  of     141976 
the  greater,  we  borrow  1  ten  from  the  first  figure  to  the  left,  Rationale, 
but  this  figure  being  0,  it  is  necessary  to  have  recourse  to 


'■li  SUBTRACTION. 

the  figure  4  of  hundreds,  from  which  1  is  borrowed,  equal 
to  10  tens  ;  but  as  only  a  single  1  is  needed,  we  leave  9  of 
them  above  the  0  ;  1  ten  is  then  added  to  5,  which  gives  15  ; 
we  then  say  9  from  15  leaves  6,  which  is  written  under  the 
units.  Proceeding  to  the  tens,  we  say  2  from  9  leaves  7.  For 
the  hundred,  as  the  upper  figure  4  has  been  diminished  by 
the  borrowed  1,  and  as  4  csnnot  be  taken  from  3,  recourse 
is  had  to  the  next  figure  on  the  left ;  but  that  and  the  figure 
to  the  left  being  the  cipher  0,  1  is- borrowed  from  the  next 
significant  figure,  3.  This  1  equals  10  of  the  order  follow, 
ing  and  100  units  of  the  order  thousands  ;  since,  however, 
we  have  need  of  only  1  unit  of  this  order,  we  leave  99  of 
tliem  which  are  placed  above  the  two  ciphers  ;  adding  1 
thousand  to  the  3  hundreds,  it  becomes  13  hundreds,  and  we 
say  4  from  ]3  leaves  9,  which  is  placed  under  the  column  of 
hundreds. 

In  the  two  following  subtractions,  each  one  of  the  ciphers 
being  replaced  by  a  9,  we  say  8  from  9  leaves  1,  and  5  from 
9  leave  4.  Passing  to  the  next,  we  say  1  from  2 — the 
figure  3  being  diminished  one — leaves  1,  or  what  is  equiva- 
lent, by  adding  what  is  borrowed  to  the  lower  figure,  2  from 
3  leaves  1. 

The  operation  could  be  thus  arranged  : 

Hundreds  of     Tens  of     mi,„„„ j„     •a„^^,.r.1^     rr ,       tt.,,^ 

thousands,    thousands.    Thousands.    Hundreds.    Tens.      Units. 

1st  num.  2  9  9  13         9       15 

2d  num.  1  5 8  4  2         9 

14  1  9  7        6 

Thus  the  greater  number  exceeds  the  less  by  6  units,  7 
tens,  9  hundreds,  1  thousand,  4  tens  of  thousands,  1  hun- 
dred thousand,  or  by  141976  units. 

Amoreconve-  Note. — The  customary  and  the  better  plan  is,  instead  of 
flientplan.        diminishing  by  one  unit  the  figure  from  which  we  borrow,  to 

leave  this  figure  unchanged,  but  augment  the  corresponding 

figure  below  by  owe  unit. 

EXERCISES    IN    ADDITION   AND  SUBTRACTION. 

Ei-ereises  on  ^7.  ExAMPLE  1. — If  a  man's  income  is  3500  dollars  a 
the  rules  of  ad- year,  and  he  spends  500  dollars  for  house  rent,  950  dollars 
traction.  for  sundry  domestic  expenses,  400,  dollars  for  travelling,  and 

350  dollars  for  charitable  purposes  :  what  will  remain  to  him 

when  the  year  is  up  ? 

2.  How  many  are  5678+7891+6035+22+9658—595? 

3.  What  is  the  sum  of  #3650+$7845+$945+$736+ 
$814+$7690-$2450? 


MVLTII'LICATION'. 


4.  How  much  are  S334+$565+$1890,  S37— 8250+ 
$6?--§20— 8(3? 

5.  A  merclKint  gains  by  tiaHiii^  at  one  time  8567,  but 
loses  by  a  bad  debt  8100  ;  at  another  time  he  pains  8075, 
but  loses  S369;  at  a  third  time  he  gains  82500,  but  loses 
$1535  :   what  remains  from  his  gains  and  Josses? 

6.  A  grocer  in  Louisville  purchased  of  a  Charleston 
house  75  hogsheads  ot  molasses.  Cov  25650  dollars,  paid 
expense  of  transportation,  365  dollars,  and  then  sold  the 
same  dn-  950  dollars  less  than  it  cost. him:  how  much  did 
he  receive  ? 

7.  If  a  trader  were  to  buy  articles  (ov  $650.  and  sell 
8250  Worth  of  them  to  one  man,  and  8125  to  another,  what 
would  be  the  amount  remaining  f(>r  sale  ? 

8.  A  farmer  bought  200  sheep,  and  gave  300  dollars  for 
them  ;  a  yoke  of  oxen,  which  cost  65  dollars  ;  a  horse,  125 
dollars  ;  a  cow,  40  dollars  ;  and  he  paid  toward  the  purchase 
100  bushels  corn,  valued  at  75  dollars  ;  200  bushels  oats,  at 
80  dollars  ;  800  weight  of  blades,  at  50  dollars,  and  gives  his 
note  for  the  balance  ;   what  is  the  amount  of  his  note  ? 

0.  If  you  purchased  100  oranges  for  250  cents,  200  limes 
tor  538  cents,  5  bunches  of  bananas  for  950  cents,  and  25 
cocoa-nuts  for  150  cents,  what  would  they  come  to  in  cents  ? 
what  in  dollars  and  cents  ?  what,  if  for  bad  fruit,  338  cents 
were  taken  off,  would  the  amount  be  ? 

10.  In  France,  during  the  Reign  of  Terror,  there  were 
guillotined,  hy  sentence  of  the  Revolutionary  Tribunal, 
nobles  1278,  women  of  the  sajiie  clas.s  760,  wives  of  artisans 
1467,  religieuses  350,  prie^its  1135,  persons  'not  noble 
13,623  ;  what  was  the  entire  number  ?  Wlnlt  is  the  dill'r. 
ence  between  this  and  the  aggregate  of  other  victims  ))ui  to 
death  in  other  forms? — women  killed  in  La  Vendee  ]5.tH)0, 
women  dying  fom  yiief  and  f(iar  3.748,  chiKIren  killed 
22.000,  men  900,000,  victims  at  Lyons  31,000,  at  Nantes 
32,000. 


iMULl'lPLICATlON. 

•■18.  Mahivlicaiion  is  a  process  by  which  is  found,  quickly,  MulUpiif'.iii'  > 
.-'-..  ',  •'  I  ^    defined, 

the  amount  ot  a  given  number.        , 

4^,  'I'he  given  nundier  is    called  iho  mulliplicand  ;  •1"'- \'",'J'^'',j'.'"^' 

nne  which  is  used  to  disc-over  the  required  amount  is  called  Multipii'nn  i 

the  mullipJicr;  and  the  answer  found  is  called  the  ■prndvct.  J*,' '"^ 

The  multiplicand  and  multiplier  are  called  factors — that  is  ) 

the  malitrs  of  the  product. 


■2-1 


MULTIFLICATIOX. 


The  sign  X-  50i  The  sign  X  placed   between  two  numbers,  denotes 

their  mvltiplicafion  togoJher  ;  the  ?e4wZ/ of  the  niuliipHca- 
pounVnumbw^' '"^^  '^^  Itnown  -is  z  compound numher  :  thus.  Gx3=18  shows, 
,  .  that  by  the  use  of  the  factors,  6  and  3,  the  result,  or  the 
compnund  number  18  has  been  found. 
duct  thoug^*^  51  •  The  product  or  result  is  invaria})ly  the  ^cr^ne,  whether 
the  figures  are  fhp  factors  stated  as  above,  or  in  the  reverse  order,  3x6. 
To  facilitate  ^^*  "^^  facilitate  multiplication,  it  is  necessary  to  keep  in 

nuiltiplication.  memory  the  sum  of  each  of  the  nine  first  numbers,  or  digits, 
•  as  they  are  called,*  repeated  from  one  time  to  nine  times  ; 
that  is,  the  products  of  each  of  the  nine  digits  by  themselves 
and  by  each  other.  The  common  tables  usually  extend 
through  12  ;  and  the  following  copies  them,  as  it  is  conve- 
"hient  to  know  more  than  the  products  of  9x9. 


MULTIPLICATION      TABLE. 


1 

2 

3 

4 

6- 

6   7 

8    9 

10   11 

12 
24 

The  inultipljea- 
tiun  table. 

2 

4 

6 

8 

10 

12 

14 

16   18 

20 

22. 

3 

6 

9 

12 

15 

18   21 

24   27 

1 

30   33 

1 

36 

4 

8 

12 

16 

20 

24   28 

32   36 

40 

44 

48  ■! 

5 

10 

15 

20 

25 

30  1  35 

40 

45 

50   55 

60 

6 

i 

12 

18 

24 

30 

36   42 

'  48   54 

60 

66 

72 
84 

1 

14 

21 

28 

36 

42   49 

56   63 

70 

77 

8 

16 

24 

32 

40 

48 

56 

04   72 

80 

88 

06 

9 

18 

27 

36 

45 

54   63 

1 
72   81 

90   99 

108 

10 
1] 

20 
22 

30 
33 

40 

50 

60   70 

80   90 

1 

100  110 

120 

44 

55 

66 

77 

88 

99 

110 

121 

13t 

12 

24 

36 

48 

60 

72 

84 

96 

108 

120 

1 
132  144 

*  So  named  from  an  old  custom  of  counting  upon  the  fingers,  and 
derived  from  the  Latin  word  digitus,  a  finger. 


MULTIPLICATION.  -0 

53.  This  table  was  made  by  writing,  as  iu  the  upper  The  muUipii.  n- 
row,  the  uumbers  1,  2,  3,  4,  otc.  The  second  by  adding  *g'","be't''^  '"^' 
these  numbers  to  themselves,  and  writing  them  directly 
under  the  first;  thus  ,1  and  1  are  2;  2  and  2  are  4 3  3 
and  3  are  '6,  etc.  The  third  row  by  adding  the  second 
to  the  first;  thus  2  and  1  are  3;  4  and  2  are  6;  0  and 
8  are  9,  etc.;  this  contains  the  first  row,  it  will  be 
observed,  three  times.  The  fourth  row  is  formed  by 
adding  the  third  row  to  the  first,  and  so  on  with  the  rest. 

5-1.  When  the  formation  of  the  table  is  comprehended,  ^,^^,  ^^^^ 
the   mode  of  use  will   be  apparent.     If,  for  instance,  the  uso. 
product  of  8  by  5,  that  is  5  times  8  were  required,  we 
look  for  8  in  the  upper  row,  then  directly  under  it,  in  the 
fifth  row,  is  found  40,  which  is  8  repeated  5  timtjs.     lu 
like  manner,  we  find  other  numbers. 

55.  To  multiply  one  number  by  another,  is  to  rc/;ea< — w"eplnui^^^^^ 
as  is  done  in  addition — ^the  first  number  as  many  times  as  a  i>rui"  ■  - 
there  arc  units,  or  ones,  iu  the  second  number.  '*  ^ '"' 

Example  1. — In  one  pasture  are  8  sheep:  how  many,,  ,,.  ,.     . 

:irc  there  in  5  pastures,  each  with  the  like  number  t  jx.rfoini,-.i  i  \ 

i;y  addition.  i'Y  multiplicatiox.         ^l]^.  ''■• 

'8 

8 

S 

s  .  8 

8  5 

Ans.        40  An^.        40 

2.  A  field  is  marked  out  into  12  tasks  :  how  many  tasks 
would  there  be  iu  12  fields  similarly  divided? 

OrKRATIO>f.  KXI'LANATJON.  E.tplar.ation  '1 

~r     ^     ■      ^^  i  11  lrl.-lr.,l  >  .1  1     t  U'  WOrK  111  til- 

:\rulliiplu-and,       12 


Multij)ly  first  by  2,  }i,s  though  given  esami«!.- 
2  were  the  only  multiplier  ;  then 
by  1,  and  set  the  first  figure  of 
the  result  in  the  place  of  tens,  or 
the  second  column,*  and  then 
the  1  by  the  1,  and  set  it  in 
third  column,  or  th:it  of  hundreds. 
Then  draw  a  line  beneath,  add  the  two  results,  and  the 
jiroduet  will  be  as  shown  in  the  cxaiiiple. 

56.  The  same  process  is  to   be  observed  whether  the 'r>'^„'f!!^? '"  ■ 
\  ample  have  few  or  many  figures. 


Multiplier,  12 

24 
12 

Product,         144 


*  Weu'e  tlie  wuici  ( iMLiniij  m  muid  rcpctiti 
id  for  Wi,nt  of  a  better  term. 


2'i  MULTIPLICATION. 

3.  lu  a  day  are  2-i  hours  :  how  many  hours  are  there  in 
28  days  ? 

OPERATION.  EXPLANATION. 


ExiManation  of 
work-. 


Multiplicand,     24 
IMultiplier,         28 


In  this  example,  we  first  say  t^ 
times  4  is  32,  and  then  place  the 
2 — a  line  being  drawn — under  the 
192  8,  or  in  the  units  column.  "We 
48  then  say  8  times  2  is  IG,  and  add 
to  that  the  3  of  the  first  multipli- 
Product,  672  cation,  which  making  19,  is  set  at 
the  left  of  the  2,  the  whole  being  192.  We  then  proceed 
to  the  second  figure  2  in  the  multiplier,  and  say  2  times  4 
(or  preferably  twice  4)  is  '8,  which  is  put  down  in  the 
•  second  or  tens  column;  and  as  nothing  remains  over  to  be 
carried,  we  proceed  to  the  nest  and  say,  2  times  (or  twice) 
2  is  .4.  This  is  placed  at  the  left  of  the  8  in  the  hun- 
dreds column,  a  line  beneath  drawn,  and'the  results  added 
for  the  product,  as  above. 
A  partial  pro-  g-j^  ']^\^q  product  obtained  by  the  multiplication  of  a 
single  figure  of  the  multiplier  is  called  a  partial  product. 
In  the  last  example,  192  and  48  are  such. 

58«  From  these  examples,  we  draw  the  following  plain 
directions  for  multiplying  a  number : 

Direction  for'  I.  Place  the  uiultiplier  under  the  multiplicand,  and  draw 
multiplying  i-    „  Kgneath 

II.  Commence  at  the  right  hand,  and  multiply  the  mul- 
tiplicand by  each  figure  in  the  multiplier,  and  place  the 
first  figure  of  each  partial  product  directly  under  its'  mul- 
tiplier, and  the  others  in  order  at  t.hc  left. 

III.  When  the  multiplication  is  completed,  draw  a  line 
beneath,  and  add  for  an  answer  the  results  of  the  multi- 
plication. 

Proof.  59,  To  prove  a  sum,  simply  reverse  the  position, of  mul- 

tiplicand and  multiplier,  and  proceed  by  the  directions. 

4.  Multiply  4320  by  345. 

5.  Multiply  58760  by  4632. 

6.  Multiply  897432  by  3G91. 

7.  Multiply  94870  by  3323. 

8.  Multiply  660761  by  4232. 

9.  Multiply  3010102  by  6654. 
10.  Multiply  34208  by  6985. 

When  ,1  cipher  ^^'  When  a  cipher  occurs  between  the  figures  of  the 
is  in  the  muiti- multiplier,  it  must,  as  much  as  anyvOther  figure,  be  set  in 
^'"■''''  its  place  in  the  partial  product. 


MULTIPLICATION. 


11.  Multiply  9567  by  8005. 


Xotc. — This   simply  sho^vs  the  value  of 
ciphers  in  certain  positions. 


0507 
8005 


47835 
'653000 


Ana.      765S3835 

12.  Multiply  90574  by  8903. 

1 3.  Multiply  79876  by  7005. 

14.  Multiply  37965  by  2002. 

15.  Multiply  54032  by  2561. 

Rkmark., — It  is  to  be'  observed,  that  0  multiplying-,  or 
multiplied  by  a  number,  5  for  instance,  is  0,  and  will,  if  it 
be  the  multiplier,  form  a  partial  product,  unless  the  briefer 
plan,  as  shown  in  the  11th  example,  be  adopted.  To  explain 
the  point  in  the  following  example,  both  methods  are  given. 
Tlic  briefer  i>s  the  better. 

16.  Multiply  3665  by  5002. 


3665 
5002 


7330 
1832500_ 

A«s.     18332330 


obbo 
51102 

J330 
0000 
0000 
18325 

Ans.     18832330 


01.  When  ciphers  are  at  the  right  hand,  the  same  can  a  oi, 
be  extended  beyond  their  usual  place,  and  simply  brought  |?j',^,^i'.' 
down,  before  we  commeuce  with  the  multiplication  of  the 
nest  figure. 

4567 
17.  Multiply  4507  by  300.  300 

Ans      1370100 


i>hf 


at   ti 
li.'iii'l. 


!-. 

Multiply 
Multiply 
Multiply 

Miiltil.lv 

90700006 
80507700 

10.  Multiply 
21.  Multiply 
23.  Multiply 

25.  :\ruitipiy 

51000504 
6530050 

20. 

69007 
2035000 

807065 
203040 

22. 

357555 
2300<i 

264 

389000 

_'l. 

3542 
765(10 

2M534 
65209000 

26.  44444x55555=? 


28 


MULTIPLICATION. 


A  way  to  short- 
en tlic  mul- 
tiplication  of 
ronipound 
iHitnTicrs. 


27.  777777x88888=? 

28.  875643x345678=? 

29.  975810x13579=? 
30.76543167X5943478=? 

62.  The  directions  already  given  can  be  used  in  all  esaiu- 
ples  of  multiplication  ;  but  .since  it  may  be  desirable  at 
times  to  shorten  the  process,  we  have  the  following  direc- 
tion :  .  - . 

03.  When  the  multiplier  is  a  compound  number,  multi- 
ply the  multiplicand  Ipj  one  of  the  factors  of  the  multi- 
plier, and  that  product  by  another  factor,  and  so  on,  until 
all  the  factors  have  been  employed.  The  last  product  will 
be  the  answer. 


F>.\iiiHple  of  foc- 
roi-jng  multipli- 
<;;ition. 


31.  Multiply  54  by  21. 

OPERATION. 
21=7X3. 

Multiplicand, 
1st  Factor  of  Mul., 

2d  Factor  of  Mul, 

Product,      ,        1134 


Remark. — The  factors  of 
21  are  the  numbers  7  and  3  ; 
and  it  is  plain  that  7  times  3 
times  a  number,  arc  21  times 
that  number. 


32.  Multiply  355  by  40,  or  its  factors  8  and  5. 

33.  Multiply  6789  by  63. 

34.  565x72=howmaDy? 

'  35.  36789 x42=how many? 
36.  96569 x36=i=how  many? 
'37.  23467 X54=how  many? 
How  to  dispose      g4^  When  ciphers  are  at  the  fight  hand  both  of  th- 
hlnlum'j'iiVand  multiplicand  and  multiplier,  it  is  sufficient  to  multiply  iu 
jiiid  multiplier.  ^^^^  ^.^^^g  already  stated,  and  then  to  place  the  full  number 
of  ciphers  at  the  right  of  the  product.      ' 

^  395 

38.  Multiply  39500  by  65000.  65 

1975 

2370 


39.  Multiply  4500  by  6500. 


Am.     2587500000 
4500 
6500 


Ans.     29250000 


MULTIPLICATION.  -•' 

10.  Multiply  345600  by  28700. 

41.  2G900X.;54300. 

42.  0785600x353200. 

43.  27000x690. 

44.  53600x70.500. 

45.  840320x543000. 

46.  2680000x135600. 

47.  901000x75300. 

TKACTICAL   EX'AJIPLES. 

I.  What  is  the  cost  of  10  pounds  of  coffee  at  15  centsa  pi'^^'J^i'luK"',: : 

pound?  '  'i  cation. 

,     2.  What  is  the  value  of  350  bushels  of  corn  at  75  cents 
a  bushel  ? 

o.  There  arc  320  rods  in  a  mile :  how  many  rods  are 
til  ere  in  265  miles  ? 

4.  Suppose  the  Charleston  Mercury  to  have  32  columns, 

and   16  paragraphs  on  an  average  in  each  column :  how 

many  paragraphs  would  there  be  in  the  paper  ? 

■      5.  Suppose  the  same  paper  to  have  9  words  in  each  line 

Vof  512  paragraphs  :  what  number  of  words  would  there  be? 

,6.  If  a  regiment  contain  780  men  :  what,  if  they  were 

viof  like  number,  would  there  be  in  120  regiments  ? 

7.  What  is  the  value  of  250  aci-es  of  land,  at  636  an  acre  ? 

8.  In  its  annual  revolution,  the  earth  moves  about  19 
miles  the  second :  how  far  does  it  move  in  a  weekjsupposing 
that  in  a  week  are  604,800  seconds? 

9.  How  many  pounds  of  cotton  in  432  bags,  each  bag 
containing  337  pounds? 

10.  Sixty-five  men  can  build  a  waJl  in  45  days  :  how  long 
would  it  take  one  n»an  to  build  a  wall  18  times  as  long? 

II.  Two  men  travel  in  diffeVeut  directions;  one  at  the 
rate  of  65  miles  a  day,  the  other  45  :  how  far  will  they 
be  apart  in  15  days  ? 

12.  What  must  I  pay  for  12  barrels  of  flour,  at  §7  a 
barrel ;  250  bushels  of  rice,  at  83  a  bushel ;  5  barrels  of 
molasses,  at  SI 8  a  barrel j  and  3  hogsheads  of  bacon,  at 
§75  a  hogshead  ?  ,       / 

Multiplication  is  simply  an  abridged  method  of  finding  ?^"'''!p''V'**'^'" 
the  sum  ot  several  equal  quantities  by  tlie  repetition  otinetiiod  to  tini 
ouc  of  those  quantities.  Hsum. 

When  the  product  of  factors,  consisting  of  several  figures, 

1    .'    .  .  u-    1  1.    c  •       .1     The  rationale  oi 

is  required,  it  is  necessary  to  multiply  each  ngure  in  thcniuitipiicatton. 
multiplicand  by  each  figure  in  the  multiplier,  and  denote 
the  several  products  in  such  order  that  they  shall  represent 
their  respective  values.     When  simple  units  a,re  employed 
I'  the  multiplier,  the  product  of  each  figure  in  themulti- 


MULtlPLICATION.  . 

piicaud  is  of  tlie  same  degree  as  the  figure  multiplied; 
that  is,  units  multiplying  units  give  units,  units  multiply- 
ing tens  give  tens,  units  multiplying  hundreds  give  hun- 
dreds, etc.  When  tens  are  employed  as  the  multiplier, 
the  product  of  each  figure  in  the  multiplicand  is  one  degree 
higher  than  the  figure  multiplied :  that  is,  tens  multiplying 
units  give  tens,  tens  multiplying  tens  give  hundreds,  tens 
multiplying  hundreds  give  thousands,  ete.  When  hundreds 
are  employed  as  the,  multiplier,  the  product  of  each  figure 
in  .'the  multiplicand  is  Uvo  degrees  higher  than  the  figure 
multiplied,  and  so  on. 

*  OPERATION. 

We  commence  by  placing      To  multiply  •  87468 

the  multiplier  under  the  mul-      By  5847 

tiplicand,  so  that  the  units  of  

the  same  order  fall  in  the  saoie  |  612276 

column.    This  being  arranged,  |  3498720 

v.'e  observe  that  to  multiply         '  69974400 

87468  by  5847,  is  to  take  the  437340000 

.multiplicand  7  times,  40  times,  

800    times,  and  5000  times;     .  511425396 

then  to  add  together  these  partial  products.  We  can  first 
find  the  product  of  87468  by  7,  which  gives  612276. 

The  second  operation  reduces  itself  to  multiplying  the 
multiplicand  by  the  figure  *4,  considered  as  expressing 
simple  units  in  writing  a  0  to  the  right  of  the  product,  and 
in  placing  the  result,  as  in  the  written  operation,  below  the 
first  partial  product.  In  like  manner,  in  order  to  perform 
the  multiplication  of  87468  by  800,  it  is  sufficient  to  mul- 
tiply that  number  by  8,  which  gives  699744,  then  .annex 
two  ciphers  to  the  right  of  -this  product ;  we  thus  have  a 
third  partial  product,  which  is  placed  below  the  two  pre- 
ceding products.  So  also  to  perform  the  multiplication  of 
87468  by  5000,  it  suffices  to  multiply  by  5,  to  annex  three 
ciphers  to  the  product  and  write  the  result,  437340000, 
below  the  first  three  products.  Adding  the  four  partial 
products,  we  have  the  total  product  511425396. 

In  the  multiplication  of  the  4,  or  the  tens  figure  in  the 
above,  we  may  conceive  that  we  have  written  one  under 
another,  40  numbers,  each  one  being  87468,  and  that  wjj 
add  them  to  obtain  the  required  product.  It  is,  however, 
evident  that  these  40  numbers  form  ten  divisions,  each  con- 
taining 4  times  87468.  Adding.and  multiplying  the  result 
by  10,  or  what  is  equivalent,  annexing  a  0,  we  obtain 
3498720  for  the  product  of  87468  by  40. 

A  similar  reasoning  applies  to  the  third  number  8;  for 


MULXirLICATION.  31 

800  numbers  equal  to  87468,  and  placed  one  under  another, ' 
form,  evidently,  100  divisions  of  8  numbers,  each  equal  to 
87408,  or  100  numbers,  equal  to  the  product  of  87468  by  8 ; 
that  is,  00!)7400  :  also,  to  the  fourth  number  5,  that  5000. 

It  is  customary  to  dispense  with  the  ciphers  to  the  riijht '*"''^  'li-ivn.-a- 
01  the  partial  product;  but  we  write  each  partial  product  of  oipiitTs: 
below  the  preceding  one,  advancing  it  one  place  to  the  left ; 
that  is,  we  make  the  first  figure  multiplied  occupy  the  same 
column  which  the  figure  by  which  we  multiply,  occupies. 

To  determine  if  an  error  has  happened  iu  the  process, 
of  multiplication,  the  following  method    of  trial,  which  er^ror!  """" 
depends  on  the  peculiar  property  of  the  number  9,  and  ,„,     .  .,.  „ 
which  is  called  casting  out  the  nines,  may  be  practised:      «f  tho  nines. 

Add  together  the  figures  of  the  product- horizontally,  i-),,,  , 
rejecting  the  number  9  as  often  as  the  sum  amounts  to 
that  number,  and  proceeding  with  the  excess,  and  finally 
denote  the  last  excess.  Perform  the  same  operation  upon 
each  of  the  factors;  theu  multiply  together  the  excesses 
of  the  factors,  and  cast  out  the  nines  from  their  product. 
If  the  excess  of  this  smaller  product  be  equal  to  the  excess 
of  the  larger  product  first  found,  the  work  may  be  supposed 
to  be  right.  Such  test  is,  however,  not  infallible;  for  if  a  » 
product  happen  to  contain  an  error  of  just  9  units  of  any 
degree,  the  excess  of  its  horizontal  sum  is  not  thereby 
altered. 

To  understand  why  the  excess  above  nines  found  in  the  The  rea.'^on 
horizontal  sum  of  a  product,  must  be  equal  to  the  excess  ^^^'^'" 
found  in  the  product  of  the  excesses  of  the  factors,  it  is  to 
be  noticed  that  by  the  law  of  notation,  a  figure  is  increased 
nine  times  its  value  by  its  removal  one  place  to  the  left; 
and  hence,  however  far  a  figure  is  removed  from  the  place 
of  units,  when  its  nines  are  excluded,  its  remainder  can 
only  be  itself  Hence,  any  number,  and  the  horizontal  sum 
of  its  figures,  must  have  equal  remainders  when  their  nines 
are  excluded.  This  being  understood,  let  it  be  observed, 
that  since  factors  composed  of  entire  nines  will  give  a 
product  consisting  of  entire  ninus,  it  follows  that  any  exce.-<s 
above  nines  in  a  product  must  arise  from  an  excess  above 
nines  in  the  factors.  TJierefore,  the  product  of  the  excesses 
of  the  factors  must  cont&in  the  same  excess  tliat  is  coutaineJ 
in  the  product  of  the  whole  factors. 


DIVISION. 


DIVISION: 

iiiviM  >;i  g5.  Division  is  a  process  used  to  find  how  many  times 

'  '^"^ '•  one  number  is  contained  in  another. 

X-  .V,  e  ^v.,.  1         66*  T^6  ^"'st  number  used  for  this  purpose  is  called  the 

-Names  employ-  ,..  ,  ,.  hiit.,,  it        ■,  •    -, 

ed  in  division,  dwisor ;  the  sccond  IS  called  the  dividend;  and  the  third, 

or  result,  is  called  the  g'MO^J'/enf.  . 

The  overplus  or      g7t  When  any  number  is  over  to  the  performed  division, 

rernaiuder.  .,   .  ii     i   .1.7-  *■  . 

it  IS  called  the  remamder. 
^■j,..^^^^,f  6S«   Three   signs    are    employed    to    express    division, 

vi-^ion.  namely,  -j-  j  —  ')  ;  and  are  thus  used,  as  in  the  case  where 

15  is  to  be  divided  by  5  ;  15-r-5  ;  -V-;  5)15. 
Wj)on  tiie  divi-     69.  When  the  divisor  exceeds  12,  and  the  divisionary 
""n^^u^ed'^"^^^  ^"^^^  )  ^^  "'^cd,  one  is  drawn  corresponding  to  it  (  on  the 

right  of  the  dividend,  and  the  quotient  placed  against  it; 

thus,  15)135(9  quoticRt. 
Division  of  two      'J^^  Division  is  distinguished  as  sJiort  and  lonff. 
^-'hort  division.       '^i*   Short  division  is  the  method  used  when  the  results 

simply  are  written,  in  consequence  of  the  divisor^,being  12. 

or  less  than  12;  for  its  performance  observe  the  following 
"  directions: 

72.  I-  Write  the  divisor  on  the  left  of  the  dividend. 
Short  division.  Begin  at  the  left,  and  having  divided  the  figure,  or  the 

fewest  figures  in  the  dividend,  that  contain  the  divisor,  set 

each  quotient  figure  under  its  dividend. 

II.  When  there  is  a  remainder  after  any  division,  annex 

to  it  the  next  figure  of  the  dividend  and  proceed  as  before- 
Ill.  Should  any  figure  of  the  dividend  be  less  than  the 

divisor,  put  down  0  for  its  quotient,  and  annex  the  next 

figure  of  the  dividend  for  a  new  dividend. 

IV.  Should  there  be  a  remainder  after  the  division  of 

the  last  figure,  set  the  divisor  under  it,  and  annex  the  result 

to  the  quotient.  .  '  • 

73.  To  discover  if  the  work  is  ciorrect,  multiply  the 
Proof.              divisor  by  the  quotient,  and  if  there-js  a  remainder  add  it 

to  the  product.     This  will  give  the  dividend  if  the  division 
has  been  correctly  performed. 

EXAMPLE    1. 

Divisor,  3)6396,  Dividend. 


Quotient,     2132 
3 

Proof,     6396 


DIVISION. 


2.  Divi(Je  t340  by  5. 

OPKHATION. 

Divisor,  5)7o40,  Dividend. 


Quotient, 


14US 
5 


7HV, 


KXPLANATION.  Explannt.on  o 

In  this  sum  we  say  5  into  w"'''*  ^^  opera 
7,  1  time  (or,  preferably/'**" 
once)  and  2  over;  setting 
down  the  1  under  the  7, 
and  annexing  the  2  that 
was  over  to  the  3,  we  say  5 
setting  down  tlie  4  under  the 


Proo 
into  2o,  4  times  and  o  over 
i5,  and  annexing  the  8  that  was  over  to  tlie  4,  we  say  5  into 
J>4.  G  times  and  4  over;  setting  the  0  under  the  4,  and 
annexing  the  4  to  the  0,  we  r>ay  5  into  40,  8  times,  which 
placed  uiicicr  the  0,  concludes  the  sum. 

3.  Divide  8JU0  by  5. 

4.  Divide  42:UU  by  4. 

5.  Divide  7G-'l     by  G. 
6    Div^lc  N()7.'n)y  7. 

7.  Divide  9") 1 80  by  l>. 

8.  Divide  67674U  by  8. 

9.  ])ivide  27840  by  5. 

No/r.—As  in  tliis  example,  the  divif^or  5  is  not  contained  xvhcm  the  dnr 
in  the  tirsl  figuie  2,  oi'  the  dividend;  w,e  say  &  into  2"7y^nd  po'e-\<^eedstb^ 
then  proceed. 

]  0.  Divide  56128456788  %  12. 

n.  Divide  4r)6780Di3:.'77  by  11. 

12.   Divide  768492Ul(/-340  by  10. 

Divide  507><0-l3i'l  by  9. 

Divide  ()470.)4.'i4  by8.— ^«.s.  {^045679  and  2  over. 

Divide  324.o<)'75  by  5. 

67^5-4^1=:'' 

7S9n4-f-5=  ? 

18.  8-.'3.")0(i-f-  =:' 

19.  5i:;70U?-H;'=? 

20.  ":-'1i;M)nf;-^<'—  ' 


first  figure  <>< 
the  dividend 


13 
14. 
15. 
16. 
17. 


i-ONG  DIVISION 

74,    \< -nil    ilio  divi.sor  exceeds   12,  ii    is  cUv'^tomary  to  when  long  div 
employ  the  method   called  Lvtif/  Divmo  .     In   this,  the '''^" '^  "^^^'^ 
entire  work   is   i)Ut  down  j    the  mental  calculation,  as  in 
short  division,  being  dispensed  with. 

I^y,  To  perform  Lt)ng  Division —  ' 

I.   Write  the   divisor  and  dividend  as  in  short  division, 
and  draw  a  curve  on  the  right  ot  the  dividend 
•  4     ■   ■  ' 


'4i  DIVISION. 

How  to  perform     II.  Divide  the  smallest  number  of  figutes  in  the  left  ol 
long  division,    j-j^g-  dividend,  that  will  contain   the  divisor,  and  set  the 

result,  as  the  first  figure  of  the  quotient;, at  the  right  of 

the  divicleud. 

III.  Multiply  the  divisor  by  each  now  quotient  figure, 

and  set  the  product  under  that  part  of  the  dividend  taken 

for  division. 
What  is  a  par-       IV.  Subtract  the  product  from  the  figures  over  it,  and 
tiai  dividend?    ^^  ^^^  remainder  annex  the  nest  figure  of  the  dividend  for 

a  new  partial  dividend. 

V.  Divide  the  partial  dividend,  and  proceed  as  before. 

until  the  whole  dividend  is  exhausted. 

21.  Divide  46755  by  15. 

OPERATION.  EXPLANATION. 

15)46755(3 117^4ns.{       In  this  example,  we  say  15  into 
Explanation  of  45  45   3  fcimey,  wliich  3  is  put  in  the 

t}w>  work  I  .  . 

—  I  quotient ;   then  we  multiply  the 

17  divisor  by  the  3,  and  set  the  result. 

15  45,  under  the  46  in  the  left  of  the 

—  dividend;  we  next  subtract  the  45 

25  ?rom  the   46,  and  having  drawn 

15  a  line  beneath  the  45,  put  "the 

—  reixainder,  1,   under  the  5,    and 

105  draw  down  the  next  figure  in  the 

105  dividend,  7;  we  then  say  15  into 

17,   1,  and  putting  this  1  in  the 

quotient,  multiply  the  divisor  by  the  1,  and  place  the  15 
under  the  17 ;  drawing  a  line  beneath,  we  subtract  the  5 
from  the  7,  and  to  the  2  annex  the  next  figure  5  in  the 
dividend,  and  so  on  until  the  division  i?  completed. 

22.  Divide  678954  by  25. 

23.  Divide  78956  by  35. 

24.  Divide  954321,  by  45. 

25.  Divide  345687  by- 50. 

When  the  divisor  contains  three  or  more  figures,  some- 
times it  is  difficult  to  discover  how  many  times  the  divisor 
is  contained  in  the  figures  separated  as  a  partial  dividend. 
Often  the  difficulty  is  obviated  or  decreased  by  trying  the 
first  figure  of  the  divisor  into  the  first  figure  or  first  two 
figures  of  the  partial  dividend.  vSuch  trial  indicates  nearh 
the  true  figure. 


DIVISION.  ;i'> 


26.  Divide  436940074  by  64237. 

OPKRATION.  EXPLWATION 

64237)436940074(6802  Ans. 
385422 


515180 
513896 


128474 

128474 


111  this  example,  we  Unci 
how  many  times  6,  the  first 
figure  of  the  divisor,  is  con- 
tiiiued  in  43,  the  first  two 
figures  at  the  left  of  the 
dividend.  It  is  found  to 
be  7,  and  7  is  contained  6 
times.     This  is  the  limit  The  iiuiit.  or  «- 

,-,         .    ,  ,,     ,  tent  m  hjvisiou- 

or  extent.  i>y  trial,  we  nud  :.ry  trial. 
it  cannot  be  7,  for  7 -times  6  are  42,  which  subtracted  fr^m 
43,  leaves  1  to  be  joined  to  the  next  figure  6,  for  a  new 
partial  dividend.  But  4,  the  s'econd  figure  of  the  divisor, 
is  not  contained  7  times  in  16,  therefore  6  and  not  7,  will 
be  the  first  figure  in  the  quotient. 

27.  Divide  56679670  by  3456. 

28.  Divide  89765132  by  94432. 

29.  Divide  7543215  by  876504. 

30.  Divide  37651245  by  508912. 

31.  Divide.  29543278  by  896302.      32.  35678-=-412=  r 
33.  56945-4-375=  ?  34.  88003-4-510=  'f 

76,  AVhen  the  divisor  is  10,  100,  1000,  etc.,  cut  from  Whm.  ^tho^diw 
the  right  hand  side  as  many  figures  as  there  are  ciphers  in  etc.        ' 
the  divisor.     The  figures  at  the  left  will  be  the  quotient, 
and  thi)se  at  the  risjht  the  remainder. 

35.  Divide  846"by  10.     Am.  S4-6,  or  84  quotient  and 
6  remainder. 

36.  Divide  95607  by  10. 

37.  Divide  74.568  by  10. 

38.  Divide  76543  by  100.     Ans.  765-43 

39.  Divide  234678  by  100. 

40.  Divide  35987654  by  1000.     Aas.  35987-654. 

41.  Divide  5976U20  by  1000. 

42.  Divide  9396781  by  1000.      -^ 

77t  VVhen  the  divisor  is  a  compound  number  (Art.  50),  a  compound  di 
a  short  method  to  find  the  quotient  is  to  separate  the  divisor  'nto^itsl-a^tors^ 
into  factors,  and  proceed  as  in  the  Ibllowing : 

43.  Divide  626800  by  40,  that  is,  by  its  factors  5x8=40. 

OPERATION.  ^^^  ...Oj. 

1st  Factor,  5)626800,    Dividend.  j. -  t-^.  v..j. 


2d    Factor,  8)125360,  1st  Quotient. 

15670,  2d  Quotient,  or  Result. 


M 


DIVISION. 


Oiviaion 
factors. 


by  3 


44.  Divide  1678  by  24. 

45.  Divide  1896  by  35. 

46.  Divide  3564  by  54. 

47.  Divide  6780  by  56. 

48.  Divide  32960  by  21. 

78«  When  the  divisor  can  be  resolved  into  3  factors  the' 
like  process  is  to  be  performed,  thus  : 

OPERATION. 


Work  perforin-     49    pjvide  990576  by  108,  or 
by  the  factors  9X6X2=108. 


9)990576 
6)110064 


2)18344 


9172   Quotient. 

50.  Divide  187236  by  252. 

51.  Divide  1255872  by  192. 

52.  Divide  12393327  by  189. 

iuv,o„  n,^..»  o..^      79.  Should  there  be  remaindei's  to  the  different  divis- 

•Tneu  sneie  are  .  1.1         ii  .1111  ^•    •  i 

remainders  to  lous,  multiply  the  hist  remainder  by  the  last  divisor  but 
i.m.  ""^^    '^'^'one,  and   add  in  the  preceding  remainder;  then  multiply 

this  result  by  the  next  preceding  divisor,  and  add  in  the 

remainder,  and   so  on  until  the  first  remainder  is  added; 

the  sum  obtained  in  this  way  will  be  the  true  remainder. 
Nute. — Should  any  remainder  be  a  cipher,  then  let  a 

cipher  be  added. 

OPERATION. 


If  the  remain- 
der is  a  cipher. 


53.  Divide  15956  by  280,  that 
is,  by  the  factors  7X5x8=280. 


7)15956 


5)2279-3  Remainder, 
8)455-4  Rem. 


Work   showing 
the  true  remain- 

lidr. 


56-7  Rem. 

True  remainder  as  seen  by  the  following  division  : 
280)15956(56 
1400 


1956 
1680 


276 
By   application    of    method  (Art.  79),   7x5+4=39; 
39x7+o=:276,  the  true  remainder. 
54.  Divide  3765452  by  126. 


i>rvisioiV. 

Oh.  543/650^-4156=  ? 

o/.   How  many  times  is  qin 

^S.   What  would  you  J  vfde  sl;^^^  '"  "^^^3  ? 
quotient  475  ?  ^''^'^  ^^7b54  by,  to  have  for  its 

59;  If  a  dividend  is  G94505  tlio  .     .- 
rerna.nder  i  19 :  what  is  the  d?;is;%'^"'*^''"^  ^^06,  and  the 

61.^)ivide  'iS^^r    ''"  ^"^^^-^  b«-ath,  t^?|f^r^. 
•578934128 
56  "20676-6  Ans. 


189 
168 

213 

196 

174 

168 


62.  Divid6  89765643  by  29 

63.  Divide  53740912  W  356. 

64.  Divide  9655468  b3  4..78. 


ticu  h.i„i  „„eV  they've,,  pTp?;     "'"""  '"'"'■  ""  9uo. 

,if  tunes  may  be  eurric!  down  toThi  I  ""V'"''  """"^^^ 
ind  divided  therewith,  wien  th  ' Til  7""  '^"'' "^  "'^'*« 
in  exact  number  of  tiles  in  the  Li  'T  'l""'  ^^"tained 
•emainder  at  the  end  c'  th  one,  f  "'h''^''''  "'"  ^^  ^ 
■  pai-t  of.  the  dividend  and  is  t?.  i  /•  •  i  \''  '•«»ia'"der  is 
ient  will  be  smaller  thn.  un  "t  "'^"^  '  ^"^  ''«  <1^°- 

!ivi<lend  just  equal  tc^he  diS  XT.  "  ,^"'"''^^  ''^' ^^ 
"'•'iont.  ^^'^  S'^es  only  a  unit  in  the 


:}8 


DIVISION. 


v.hy  in  division      In  the  first  three  operations  of  arithmetic,  the- filcu 
thcwtrk  i.j.  be-ti  ^  performed  by  commeucing  at  the  P^rtit;  in  «i>i- 

,unatt.eief,.  ^.^^  ^^^  .J^.^ence  at  the  left,  because  .^hc  dundend  being 
the  sum  of  the  partial  products  of  the  divisor,  by  the  units, 
tens,  hundreds,  etc  .of  the  quotient,  all  these  partial  pro- 
ducts are  luin^'le'l  ^"^  ^'*''  another,  so  that  it  is  impossible 
to  eommencoV^sf  ^^'•fi"^  out  the  product  by  the  units, 
V     ,vg  ^^,„.^,  et<r.     By  the  established  method,  we  are  able 
^^Jj^.oimin'fe  at  once  in  what  part  of  the  dividend  the  pro- 
duct by  the  highest  units  is  found,  and  then  we  obtain  the 
figure  of  the  highc.'-t  units,  and  then  arrive  at  the  figure  of 
the  units'of  the  order  immediately  below  the  first,  and  so  on. 
;^„.i-.uhPn      When  the  dividend,  divisor  and  quotient  contain  any 
any  number  of  Quj^jjcf  of  .fioures,  as  in  the  accompanying  sum,  the  lollow- 
ing  explanation  IS  given  : 

Placing  mentally  three  ?eros,and-afteN 
wards  four  zeros  to  the  right  of  the  divi- 
sor, we  obtain  two  products,  2(i78U00,  and 
2G780000,  the  one  less,  the  other  greater 
than  the  dividend.     Thus,  the  total  C|U0- 
tient    is    com])nsed   between    1000  and 
'  10,000,or  must  be  composed  of- four  fig- 
ures, of  which  the  first  to  the  left  ex- 
presses thomamh.     In  order  to  find  this 
first  figure,  we  observe  that  its  pibduct  by  the  divi.'-or,  inas- 
much "as  it  is  thousands,  is  to  bd found  wholly  ip  the  part 
917(5  thousands  of  the  dividend.  I  We  are  then  led  to  divide 
9176,  which  we  consider  as  a  firstipartial  dividend,  by  '^078 ; 
and  the  gitatest  number  of  timet  that  the  second  number 
is  contained  in  the  first  represenW  the  tli(Ai&uv(h  figure  of 
the  whole  quotient.     Now,  the  \rue  quotient  of  9176  by 
2678,  obtained  according  to  the  i\efhod  of  trial  indicated, 
is  3.     We  write,  then,  3  below  the'<|visor.     Next,  we  sub- 
tract from    the  dividend  tbe  produ:t  of  the  divisor  by  8, 
either  by  placing  thjs  product  belo'  the  partial  dividend, 
and  subtracting  or  efiecting  simu'' aneously  the  f-ubtr, 
tion  and  the  multiplication  as  abov .     Ihis  first  oi-erati 
amounts,  evidently,  to  sublractipg  000  times  the  divisor 
from  the  dividend.  -  . 

The  remainder  of  the  first  subtration  being  1142,  if  wqI 
write  after  it  the  figures  of  the  di  !dend,  which  have  not 
yet  been  used,  there  would  resultla  new  dividend,  upon 
which  we  could  operate  as  upon  th  first  dividend;  but  ae 
we  have  now  to  determine  the  hunceds  figure  of  the  quo- 
tient, and  as  the  product  of  .the  di\ior  by  this  figure  ca 


91762981:^678 
11422     342'6 

7109 
17538 

1470 


DIVISION. 


M^ 


not  give  units  of  a  lower  order  than  hundreds,  it  must  be 
contained  wholly  in  the  11422  hundreds  of  the  remaining 
dividend;  so  \vo  hrinii^down  to  the  right  of  the  remainder, 
■1142,  only  the  follo\Tlng  iijiure  2  of  the  dividend,  which 
gives  a  second  partial  dividend,  11  •422,'  upon  which  we 
operate  as  on  the  first.  .  * 

The  true  quotient  of  11422  by  2078,18  4,  which  we 
write  below  the  divisor,  and  to  the  right  of  the  first  quo- 
tient ubtiiined.  We  then  subtract  from  the  second  partial 
dividend,  the  product  of  the  divisor  by  the  new  (juotient. 
The  remainder  of  this  subtraction  is  710.  We  bringdown  . 
to  its  right  the  following  figure  of  the  dividend,  9,  which 
gives  a  third  dividend,  71 00,  and  which  is  to  furnish  the 
t<.ms  figure  of  the  whole  quotient. 

Dividing  7109  by  2078,  we  have  for  a  true  quotient  2, 
which  we  place  at  the  right. of  the  first  two  figures  of  the 
quotient;  multiplying  the  divisor  by  2,  and  subtracting  the 
product  from  the  third  partial  dividend,  we  obtain  1753 
for  a  remainder,  to  the  rigbt  of  which  we  bring  down  the 
last  fiLiure  8  of  the  dividend,  which  gives  1753S  for  a  fourth 
pjirtial  dividend.  Finally,  the  true  quotient  of  17588  by 
267«^,  is  G.  We  multiply  the  divisor  by  0,  and  subtract  the 
product  from  the  fourth  partial  dividend,  which  gives  a 
remainder,  1470.  The  required  quotient  is  then  3426, 
with  the  remainder  1470,  which  we  can  prove  by  multiply- 
ing 2U7.8  by  3426,  and  adding  1470  to  the  product.  The 
four  operations  just  performed  in  this  division,  conduct  to 
the  same  result  as  if  we  had  successively  subtracted  from 
the  dividend  3000  times,  then  400  times,  then  20  times, 
then  G  times  the  pro]>osed  divisor. 

.  Since  in  division,  the  dividend  is  a  product  of  which  thecert»iu  truths 
divisor  and  quotient  are  two  factors,  it  follows  that  to  divide 
•the  dividend  by  a  certain  entire  number,  the  quotient  is  by 
this  change  divided  by  the  same  entire  number.  .  For,  as, 
after  this  change,  the  quotient  multiplied  by  "the  divisor 
must  produce  a  dividend  a  certain  nuuiberof  times  greater 
or  less  than  the  first  dividend,  it  follows  necessarily,  the 
divisor  remaining  the  same,  that  the  quotient  must  be  the 
>' '"le  number  of  times  greater  or  less.- 

)n   the   contrary,  if,  without 'altering  the  dividend,  we 

under  the  divisor  a  certain   number  of  times  greater  or 

smaller,  the  quotient  is  thereby  rendered  the  same  number 

of  times  smaller  or  greater.     Then  by  dividing  the  divi- 

,  dend  and  the  divisor  by  the  same  number,  we  do  not  change 

;£lie  quotieat;  since,  if,  by  the  change  of  the  dividend,  we 


^*^  MlSCELLANEOtS    EXABIPLKS. 

divide  the  quotient  by  a  certain  number,  the  second  cbaugt 
renders  it  the  same  number  of  times  smaller  or  greater. 
Thus,  the  compensation  leaves  it  the  sami-. 


MISCELLANEOUS   EXAMPLES    ON   .THE 
INTRO.DUCTOEY  PART. 


SECTION    I. 


Kxamplos  in 
Votation, 


Siiditjo)!. 


."■'jilitniofion. 


tfultiplicaiion 
ind  Division. 


Example  1. — Write  in  Koman  Notation  9,  99,  100, 
1000,  2060. 

2.  Write  in  figures  six,  sixty,  six  hundred,  ono 
thousand,  nine  liuiidrcd. 

3.  Write  in  figures  three  trillions,  t>wo  hundred  and 
forty  billions  and  five. 

4.  Write  in  words  35,644,760,027,201.   • 

5.  Add  500,  650,  4709,  810,  80,  1,  and  29  together. 

6.  Add  2,  20,  35,  595,  6a6,  890,  and  5  together. 

7.  From  one  hundred  take  ten. 

8.  From  two  hundred  and  fi^fty  take  eighty. 

9.  From  one  hundred  take  one. 

10.  From  four  hundred  and  sixty-two  take  sixty- 
five. 

11.  Add  5000,  659,  7005,  89,  and  100  together.      , 

12.  Add  99,  990.  709,  56,  and  47  together. 

18.  How  many  times  are  483  contained  in  568943!' 

14.  How  many  times  are  6340  contained  in  2076907  ''. 

15.  Howmanvtimesai-eSoOOcontainedin 37695843? 

16.  Add  55+664-706+94+2  together: 

17.  Subtract  56785  from  1578790. 

18.  9340760—65007=  ? 

19.  725631—483...=  ? 

20.  What  is  the  difference  between  1776  and  1861? 

21.  What  is  the  product  of  five  thousand  eight 
hundred  and  seventy, by  two  hundred  and  sixty-five? 

22.  What  is  the  product  of  thirty-nine  thousand 
seven  hundred  and  thirty-six,  by  eight  hundred  and 
twenty-five  ? 


MISCE^iLANEOUS    ESAMPLES.  -i' 

23.'  Subtract  4596006  from  58560C7('.  Example  in 

24.  How  many  times  arc  64:Z  (iontaiiicd  in  1000  ?      PubtSon. 

25.  Howmany  times  arc  71o2  coiitiuncd  in  10000?    MuitipHc-.toou 

26.  IIow  many  times  are  ;»SU:i  contained  in  1000000  ?  pivisio-. 

27.  Subtract  U)056784  from  875648201. 

28.  68541 — 3954  arc  how  many? 

29.  Twenty  thousand  times  thirty  thousand  are  how 
many  ? 

30.  Sixty-five  thousand  times  five  thousand  and 
dighty-ibur  arc  how  many  ? 

31.  What  is  the  product  of  eiglity-onc  thousand  Iwo 
hundred  and  seven,  by  three  thousand  one  hundred 
and  forty-five  ? 

32.  >yhat  is  the  product  of  thirty-seven  thousand 
five  hundred  and  sixty -five,  by  two  thousand  and 
fifty-two  ?  j 

83.  From  four  thousand  four  liundred  and  tvventy- 
tiinc,  take  two  thousand  and  sixty-eight.  / 

34.  Front  five  thousand  seven  hundred  and  s^vehty- 
Jioven,  take  six  hundred  and  eighty-lour. 

35.  Multiply  ten  millions  by  ten. 

36.  Multiply  thirty  millions  bj'  one  hundred. 

37.  Multipl}'  one  thousand  by  one  thousand.  . 

38.  From  one  hundred  thousand  take  one.     " 

39.  65-|-405+76y-f8905,  are  how  many? 

40.  3695-^340,  are  how  many? 

41.  785U6x695,  are  how  nmny? 

42.  597092 — 579,  are  how  many  ? 

43.  How  many  times  are  5684  contained  in 
16784320  ? 

44.  How  many  times  are  1804  contained  in  286784? 

45.  How  miiny  times  are  695  contained  in  457803? 

46.  From  one  thousand  take  nine  hundred  and 
ninetj'-nine. 

47.  Multiply  seven  millions  six  thousiad  and  thirty 
bv  thirty -three. 

*48.  Divide  87658910  by  795. 

49.  Divide  59374390  by  622. 

50.  Divide  38427695  by  3652. 

51.  Multiply  9867846  bv  5890. 


Ri'DUCTION. 


•    EKDUCTION. 

COMPOUND     NUMBERS. 
SECTION   II. 

Uoduciuivi  de-       ^^*  i^^duction  is  tiio  change  of  numbers  from  one 
uw.d.  name  or  denomination  to  that  of  another,  but  without 

change  of  value. 
'!y«  numi-.erij      82.  When  numbers  are  to  be  changed  fiom  a  higher 
io''d1fferlnf     ^^  ^  lov/er  name,  mv.ltiplication  is  employed;  but  when 
'i-tnis.  tlie  change  is  from  a  lower  to  a  higher  one,  we  use 

division. 
<>>mt)ounu^  83.  I^umberB  subject   to    such    changes  are  called 

"ul!}eHiJ^ih^L^^'^^P^''^^^d,m  distinction  to  t\\esi77i2)le  numbers  already 
■<.hc..nges.  Considered. 

A  compuiind        84.  A  compouncl  number  is  made  of  two  or  more 
11"^^^'"'^*^;      unlike    denominations;    thus    dollars,   cents,   dimes, 

pounds,  shillings  and  pence,  are  of  this  cluss. 
What  are  not       85.  It  should  be  notcd,  that  while  the  sevcr.il  parts 
mimbers"^        °^  '^  compound  number,  as  pounds,  shillings-!,  pence, 
are  of  different  name,  they  are  classed  together  as 
relatively  alike  ;  but  pounds  and  dollars,  and  grains  and 
minutes,  etc.,   having   no  common  bond,  cannot  be 
reckoned  as  compound  numbers.     All  of  such  nature 
are  expressions  of  unlike  values. 
HoH  to  ppduoe     86.  To  reducc  a  compound  number  to  one  of  a  lower 
r°»^!r''^         name,  observe  the  following  directions  : 

Multiply  the  highest  denomination  in  the  number  to 
be  changed,  by  that  figure  which  indicates,  how  mapy 
ones  or  units  of  the  next  lower  denomination  are  con- 
•    tained  in  one  of  the  highest,  and  add  to  the  product 
the  parts  of  the  same  value  with  the  multiplier,  and 
80  on. 
rabie  of  values     87.  The  following  tables  of  values  should  be  thor- 
loi  memory,     oughly  memorized.     The  tables  of  refef-ence  beneath 
TaWes  for  refer- them    arc  not  to  bc  Studied,  but  used  for  comparing 
work  done  by  the  pupil.     They  simply  show,  in  a  con- 
densed form,  the  tables  of  values,  thus: 
1  £=20.s.=240(?.=960gr.  or  farthings. 


REDUCTION^ 


4.-i 


ENGLISH    MONEY. 


S8-     4  Farthings,  marked  qr.*  make  I  penny,  d. 
12  Pence  make  1  slnliing, ,«. 

20  F^hillings  make  1  potind,  £. 

21  Shillin!L!;s  make  1  i>innea: 


Tuble  of  Km 
crlish  moni^v 


Reference  Tabic. 

S.  '         d. 

1 

1  =  12 

s:0  =       24(1 


=  4S 

=       0(U) 


Example  1 - 
operation. 


10 


20 

lis 


12 

8 


21.5.?. 
12 

2588rf. 
4 


-E(*diice  £10  15s.  M.  2qr:  to  farthing^.; 

EXPLANATION. 


Ten  poiinds=200s.,  and  the  ^:^,?^|-"^'' 
155.  added  make  215s.;  thi.s 
number  multiplied  by  12=2580, 
iiiid  with  the  8^/.  added,  2588  ; 
this  multi)died  by  4=10:352, 
and  the  2  added,  10854,  which 
is  tlie  answer. 


Woi)\qr.  Ans.  \ 

2.  Reduce  £2518.  Gd.  3q}\  to  lUrthings. 

^otc- — It  is  convenient  to  place  in  small  figures,  as 
in  the  first  example,  above  the  numbers  of  the  sum, 
the  several  multipliers. 


o.  Reduce  £37  16s!  f>d.  to  pence. 

4.  Reduce  £l9  Os.  4^^.  to  farthings. 

5.  Reduce  £40 -85.  5.Vrf.  to  farthings. 

6.  Reduce  £32  3s.  9d.  to  pence. 

7.  Rc<luce  £50  lO.s.  to  shillings. 

8.  Reduce  £42  15s.  Od.  to  ponce. 
0.  Reduce  £543  Os.  'dd.  to  pence. 

10.  Reduce  £80  Is.  2d.  to  pence. 

11.  lU'duce  £(>  6s.  Qd.  to  pence. 

12.  Reduce  £48  12s.  6.^Z.  to  farthings. 

13.  In  5  guineas,  how  many  pence  are  there 


Kxamples  in 
Knglisn  mnii(> 


Qr.  is  quarter,  the  quart'.^r  or  fourth  part  of  a  penny. 


My  iMfysure. 


■'i-^  REDUCTION. 

14.  Eod^^ce  50  guineas  to  shillings. 
16.  Eeduce  2  guineas  to  shillings. 

IS^ote. — Whenever  there  is  omission  of  any  term,  as 
that  of  shillings  in  the  4th  sum,  it  does  not  afteet  the 
multiplication  by  shillings;  thus  19  is  to  be  multiplied 
by  20,  though  the  order  of  shillings  is  not  i!i  the  sun:. 

DRY    MEASURE. 

8Si  2  Pints,  pt.  make  1  quart,  qt. 
8  Quai-ts  make  1  peck,  pk. 
4  Pecks  make  1  bushel,  bush. 
36  Bushels  make  1  chaldron,  ch. 


bush. 


Note. — By  this  table,  all  dry  articles,  as  grain,  salt, 
coal,  vegetables,  etc.,  are  measured. 

Example  1. — How  many   pints  are   there   in    1/ 
bushels,  3. jiecks,  5  quarts  and  1  pint? 

4  8  ^2 

lo  bush.  Spk.  b  qt.  \  pt. 

4 

m  pk. 

8 


509  qt. 


Reference 

Table. 

pk. 

qt. 

pt. 

1 

= 

2 

1            = 

8 

= 

16 

4         = 

32 

= 

64 

.4715.     1020  pt. 


k.j.iuripies  m  '^-  ^^luce  10  bushels,  3  pecks,  5  quarts  to  pints. 

ih-v  i,ie;.,s-are.  8.  Eeduco  25  bushcIs,  2  pecks,  6  quarts  to  quartp. 

4.  How  many  pecks  in  35  bushels  ? 

5.  How  many  pints  in  28  quarts  ? 

6.  How  many  pints  in  3  pecks  and  6  quarts? 

7.  How  many  quarts  in  2  bushels  and  2  pecks? 

8.  Eeduce  144  chaldrons  to  bushels. 

9.  Eeduce  46  chaldrons  to  pecks. 

10.  Eeduce  25  bushels,  3  pecks,  7  quarts,  1  pint,  tc' 
•  pints. 


KEDUCTION. 

11.  Eeduce  75  bushels  and  2  pecks  to  quarts. 

12.  Eeducb  250  Sushels  to  pecks. 

.LIQUID    MEASURE. 

90.     -i     Gills,  gi.  make  1  pint,  pt.  ■  -:q«' 

2     Pints  make  1  quart,  qt. 
4     (Quarts  make  1  gallon,  gal. 
)\\l   Gallons  make  1  barrel,  6^*/. 
Go     Galloiis  (iil.]x2)  make  1' hogshead,  hhd. 
2     llogsheadiS  make  1  pipe,/)/. 
2     Pipes  or  4  hogsheads,  make  1  tun,  tun. 
The  tierce  in  tables  called  42  gallons,  is  omitted. 
'X»  it  does  not  represout  the  tierce  used  in  trade,  which 
baa  sometimes  many  moi*e  gallons. 

lieferctice   Table, 
hin.     pi.     hhd.        bbl.         gal.  qt.       pt.       gi.  • 

1=       4 

1  =       4=      8=     ^2 

I     =  SU=  126=  252=U)UB 

1     =     0     =  (>3  =  25.=^  504=201(5 

1   =  2     =     0     =126  =  5o4=luU8=40;52 

1    =    2  =  4     =     0     =2.")2  =lOOc^=::ul(!=8U64 

B}'  this  are  measured  all  liquids,  except  milk,  ale 

and  l)ocr. 

Tiie  Confederate  States  gallon  of  liquid  n\easure  i-i 
231  cubic  inches. 

Example  1.— Ill  6  pipes',  3  hogsheads,  15  gallona 
'.  an4  3  quarts,  how  many  quarts  ? 
"^  2    '  63  4 

0  pi.         3  hhd.  15  gal.  ?>  qt. 


15 
63 


iu;o 

4 

Ans.     3843  qt-^ 


46  .  KEDUCTION. 

Eemark. — When  a  number  to  be  added  consists  of 
two  figures,  as  the  15- in  this  sum,  it  is  more  con- 
venient to  place  as  above,  than  to  use  it  as  we  do  sin- 
gle numbers. 

2.  Eeduce  15  pipes,  1  hogshead,  3  gallons  to  quarts. 

3.  Reduce  1  tun,  30  gallons,  2  quarts,  2  pints,  •:■■ 
gills  to  gills. 

4r.  Reduce  1  hogshead,  15  gallons,  3  quarts  to  pint?. 

5.  Reduce  1  barrel,  2  quarts  to  quarts. 

4 
I  bbl.     =     ol^  gal.     2  qt.  i         explanation. 
4  _  j      ■ 

In  this  example,  a!< 

124  I  the  f  gallon=2  quarts. 

2  {   we  simply  add  i  of  4 

j  to  the  quart,  one-half 

126  !  gallon  beino-  2  quarts. 

2 

A /IS.     128  qts. 

6.  Reduce  1  hogshead,  1  barrel,  3  quarts,  2  pints  to 
pints. 

7.  Reduce  1  barrel,  3  quarts  to  gills. 

8.  Reduce  1  hogshead,  15  gallons,  3  quarts,  2  pints. 
3  gills  to  gills. 

9.  In  3  hogsheads,  12  gallons,  3  quarts,  how  many 
quarts  ? 

10.  In  one  pipe,  1  hogshead,  5  quarts,  how  many 
quarts  ? 

11.  In  1  tun,  1  barrel,  how  many  pints  ? 

12.  In  2  hogsheads,  14  gallons,  how  many  gallons'' 

AVOIRDUPOIS   WEIGHT. 

iipf"ht"^°"  SI*       1*>  I>i'aehms,  dr.  make  1  ounce,  o~. 

16  Ounces  make  1  pound,  Ih. 
25  Pounds  make  1  quarter,  qr. 
4  Quarters  make  1  hundred  weigiit,  not. 
20  Hundred  weight  make  1  ton,  t. 

Reference    Table, 

t.          act.         qr.            lb.            oz.  dr. 

1  =  16 

1  =    16  =  256 

1  =   i>5  =   400  =  6400 

1  =  4  =  100  =  1600  =  25600 

I  =  20  =  80  =  2000  =  32000  =  512000 


REDUCTION. 


Such  articles  as  sugar,  coffee^  tea,  cotton  and 
metals,  with  the  exception  of  gold  (ind  silver,  arc 
weighed  by  this  weight.  .1 

In  the  old  tables,  liS  lbs.  was  called  a  qr.,  and  112 
lbs.  a  cwt.;  but  now  the  standard  qi*.  is  25  lbs.,  and 
the  cwt.  100  lbs. 

Example  1. — Reduce  17  tons,  8  huiidrcd  weight,  -^ 
qiiarters,  13  pounds,  to  pounds. 

.    20  4  2.') 

nt/:..^cwt.   s  qr    la //>. 

20 

i'AS  cw  1 . 
4 


1395  qr. 
25 


6975 
2790 
18 


.ins.     34888  lbs. 

2.  Reduce  6t.  7cwt.  2qr.  2010.,  to  oz. 
8.  Reduce  25t.  5cwt.  Iqr.  lOlb.  2oz.,  to  oz 

4.  Reduco  181b.  lloz.  12dr.,  to  dr. 

5.  Reduce  2t.  15oz.  14dr.,  to  dr. 

6.  Reduce  Oqi-.  171b.  looz.  5dr.,  to  dr. 

7.  Reduce  3t.  22  lb.,  to  oz. 

8.  Reduce  8qr.  15oz.,  to  dr. 

9.  In  ocwt.  201b.,  how  many  pounds  ? 
10.  In  9qr.  121b.  5oz.,  how  many- ounces? 

apothecaries'  weight. 
92,  20  Grains,  </r.  make  one  scruple,  9.  Apot,hoc»iv 

3  Scruples  make  1  drachm,  5.  wojirhi. 

8  Drachms  make  1  ounce,  5. 
12  Ounces  make  1  pound,  Iti. 


Rrfcn 


lb. 


-..              ((r. 

vr. 

fr- 

I 

= 

20 

1 

=       3 

= 

t>0 

1     =     8 

=     24 

=s 

480 

12     =-96 

=  288 

= 

5760 

4S  REDUCTION. 

Medicines  are  compounded  by  this  weight,  but  bought 
and  sold  by  Avoirdupois  weight 

There  is  no  difference  in  the  pound,  ounce  and  grain 
of  this  and  Troy  weight,  but  the  ounce  is  differently  sub- 
divided.    ' 

Example  1. — Eeduce  6  pounds,  7  ounces,  5  drachms, 
2  scruples,  12  grains,  to  grains. 

'12  8  3  20 

6tb  7s  53  29  12gr. 

71) 

8 

637 
8 


■:;^^t, 


1913 
20 

Ans.        38272  f/r. 

2.  Reduce  25rb.  8oz.  2dr.  15se.,  to  scruples. 

3.  Reduce  801b.  9oz.  Idr.,  to  drachms. 

4.  Reduce  21b.  3oz.  2dr.,  to  scruples. 

5.  in  oOfb.  8oz.  2dr.  12sc.,  how  many  scruples? 

6.  In  9oz.  5dr.,  how  many  drachms? 

7.  In  50tb.  4oz.  Idr.,  how  many  scruples? 

8.  In  121b.  2dr.,  how  many  scruples  ? 

9.  In  291b.  6oz.  Idr.  IGsc,  how  many  grains? 
10.  In  5oz.  2dr.  15sc.,  how  many  grains?  ' 

TROY  WEIGHT. 

93.  24  Grains,  gr.  make  1  pennyweight,  ^j?o^ 
20  Pennyweights  make  1  ounce,  oz. 
12  Ounces  make  1  pound,  lb. 

Reference  Table. 

lb.             oz.            pwt.  gr. 

1    =  24 

1     =       20     =  4H0 
1     =     12    =     240    =     5760 
This  is  the  standard  measure  for  gold,  silver,  jewels, 
corn,  bread  and  liquors. 


REDUCTION.  41) 

Example  1. — Reduce  15  pounds,  9  ounces,  14  penny- 
weights, 12  grains,  to  grains.- 

12         20  24 

151b.     9oz.     14pwt.     12  gr. 
12 

189  oz. 
20 

3794  pwt. 
24      . 


Ans.       51068  gr. 

2.  Reduce  211b.  9oz.  J4pwt,.12gr.,  to  grains. 

3.  Reduce  5oz.  14pwt.,  to  pennyweights. 

4.  Reduce  551b.  lOoz.,  to  grains. 

5.  Reduce  9oz.  15pwt.  15gr.,  to  grains. 
■C.  Reduce  IGpwt.  6gr.,  to  grains. 

7.  Reduce  101b.  6oz.,  to -pennyweights. 

8.  In  131b.  IGpwt.  logr.,  how  many  grains? 

9.  In  351b.  14pwt.  lOgr.,  how  many  grains? 
10.  In  601b.  how  many  pennyweights? 


ALE  OR  BEER  MEASURE, 


^4,      2  Pints,  j)t.  make  1  quart,  qt.  A»e  or  beer 

4  Quarts  make  1  gallon,  ffai. 
36  Gallons  make  1  barrel,  bar. 
54  Gallons  make  1  hogshead,  hhd. 


Refo'encc  Table. 

hhiL            bar.         gal.               qt.  pt. 

1  =  2 

I     =         4  =  8 

1     ==     36     =     144  =  288 

1     =       1]  =     54     =     216  =  432 

1  Gallon  contains  282  cubic  inches.     ' 
5 


50 


REDtiCTlON. 


Example  1.— Reduce  25  hogsheads,  3  quarts,  2  piiite^ 
to  pints. 

4  2 

25hhd.     Sqt     2pL 
54 


Cloth  measure. 


100 
125 

1350  gal. 
4 


5403  qt. 
2 


Jfote. — Though  no 
gallons  are  named  in 
the  sum,  we  still  have 
to  multiply  by  the 
number  of  gallons  that ' 
make  a  hogshead. 


A71S.    10808  pt- 


2.  Reduce  5hhd,  2bar.  3qt.  Ipt.,  to  pints, 

3.  Reduce  30hhd.  3bar.  2pt.,  to  pints. 

4.  Reduce  15hhd.  15gal.  3  qt.,  to  quarts, 

5.  Reduce  12hhd.  20gal.  2qt.,  to  pints. 

6.  Reduce  25gal.,  to  pints. 

7.  Reduce  Ibar.  3qt.,  to  pints. 

8.  In  8hhd.,  how  many  quarts  ? 


95. 


2i* 

4 

4 


CLO^fH  MEASURE, 

Inches,  m.  make  1  nail,  na. 
Nails  make  1  quarter,  qr. 
Quarters  make  1  yard,  i/d. 
Quarters  make  1  Ell  Flemish,  E.  Fl 
Quarters  make  1  Ell  English,  E.  Er 


E.  Eng. 


Reference  Table, 
yd.  E.  FL  qr. 


1 
li 


1 
If 


na. 
1 
4 

12 

16 
20 


in. 

2( 

9 
27 
/]6 
45 


Cloths,  carpets  and  all  articles  of  the  yard  ineusurement 
are  sold  by  this. 


*  The  fractional  \,  and  other  like  expressions  in  thu  Tables^, 
will  be  explained  under  the  head  of  fractions/ 


REDUCTION.  51 

Example  1. — Reduce  36  yards,  3  quarters,  2  nails,  to 
nails. 

4  4 

36  yd.     3qr.     2na. 
4 

147  qr. 
4 

Ans.   590  na. 

2.  Reduce  2.5yd.  2qr.,  to  nails. 

3.  Reduce  350yd.  3qr.  3na.,  to  nails. 

4.  Reduce  3qr.  3na.,  to  nails. 

5.  Reduce  12yd.,  to  quarters. 

6.  In  28E.  Fl.,  how  many  nails? 

.     7.  In  17E.  Eug.,  how  many  quarters? 

8.  In  60E.  Fl.,  how  many  yards? 

9.  In  37E.  Eng.,  how  many  yards? 
10.  In  45yd.  3qr.,  how  many  quarters  ? 

LONG  MEASURE, 

96,     3     Barleycorns,  i.e.  make  1  inch, //?.  j.ons,' mea^wn. 

12     Inches  make  I  foot,  ft. 
3     Feet  make  1  yard,  i/d. 
5 J  Yards,  or  (5:1x3)  16^  feet,  make  1  rod,"r(?. 
40     Rods  make  1  furlong,  /ur. 
8     Furlongs  make  1  mile,  mi. 
3     jMiles  make  1  league,  l. 
'69j^  Statute  miles,  nearly,  or  60  geographical  miles, 
make    1    degree   or   circumference   of   the 
earth,  c?e</  or  ° 
360    Degrees  make  1  circumference,  cur. 


mi.    fur. 


1  = 

1  =  8  = 

By  this  is  measured  distances,  lengths,  breadths,  etc. 
A  fathom  is  a,  length  of  0  feet,  and  is  used,  principally,  j^  f,,t, ,£,, 
for  soundings,  at  sea. 


Reference  Tabic. 

rd. 

yd. 

ft.                in.              I.e. 

1  =            3 

1  =         12  =          36 

1  = 

3  =        36  =         108 

1  = 

51  = 

16. J  =       198  =         594 

40  = 

220  = 

660  =     7920  =     23760 

320  = 

1760  = 

5280  =  63360  =  190080 

A  liaiid. 


To   multiply 


REDUCTION. 

A  "hand  is  4  inches,  aud  is  used  to  find  tlie  height  of 
horses. 

To  multiply  by  J,  as  in  the  Rod  measure,  we  simply 
take  one-half  of  the  number  to  be  multiplied,  and  add; 
thus  8x5^=40x4  (or  the  half  of  8)  =44. 

T.i„  -I     .p^educe  4  ^o^^^  2  feet,  8  inches,  to  barley- 


Example  1.- 


4rd. 
5J 

20 

22  yd. 
_3 

68  ft. 
12 
824  in. 


3 
2  ft. 


12 


Ans.  2472  b.c. 


Here  it  will  be  seen, 
that  although  not  named 
in  the  sum,  the  rods  have, 
first,  to  be  changed  to 
yards  before  they  can  bo 
reduced  to  feet.  The  2 
added  to  the  20  is  the  \ 
of  4  rods. 


2.  Reduce  lOmi.  6fur.  25rd.,  to  feet. 

3.  Reduce  25°  30mi.  5fur.  lOrd.  3yd.  2ft.,  to  inches. 

4.  Reduce  50mi.  6fur.  Srd.  5yd.,  to  inches. 

5.  Reduce  5mi.  5rd.  ^ft.,  to  feet. 

6.  In  95°  how  many  miles? 

7.  In  5fur.  6rd.  9yd.  7in.,  how  many  inches? 

8.  In  8rd.  9yd.  3qr.  2ft.,  how  many  feet  ? 


rjiuifc   lf,c;i«- 


LAND  OR  SQUARE  MEASURE.* 

97«  144     Square  inches,  &€[.  m.,  make  1  sqttare  foot,  aq./t. 
9  •  Square  feet  make  1  square  yard,  sq.  yd. 
30J  Square  yards,  or  272}  square  feet  make  1 

square  rod,  sq.  rd. 
40     Square  rods  make  1  rood,  r. 
4     Roods  make  1  acre,  a. 
640     Acresf  make  1  square  mile,  sq.  mi. 


A   ?f)ti.are  de- 


■^  A  square  is  a  figure  bounded  by  four  equal  lines  at  right 
angles  to  each  other.  Each  line  is  known  as  a  side  of  tha 
square. 

t  An  acre  contains  4840  square  yards.  For  larger  areas,  there 
'  is  the  square,  one  of  the  sides  of  which  is  a  mile.  The  squaro 
mile,  containing  640  acres,  is  called  a  section  in  the  classification 
of  public  lands. 


sq.  ■)m,  a 


REDUCTION. 

Reference  TahU. 

a.           T.         sq.  rd. 

.^y.  yd. 

sq.  ft. 

87.  i)i. 

1  =. 

144 

1   — 

0  =- 

120(i 

1— 

30]  «= 

2721  = 

30204 

1=        40— 

11210  =- 

10800  = 

1 5081 GO 

1==         4=       1G0= 

4840  = 

4:]r,c,0  =. 

()272G40 

040=-  2<700«=102400=. 

;K)97000  =»2 

-878400  = 

4O1418",)()0O 

Surfaces  are  measured  by  this. 

A  square  yard  contains  9  square  feet;  the  prtxluct  of 
3x3.  A  square  foot  contains  144  square  inches,  the 
product  of  12x12. 

The  multiplication  by  the  fractional  figure  \,  as  iu  the  MiiUipii/^nur.n 
square    rod,  is  performed    by  taking  one-fourth  of  the t*i.iu'"'.'"" 
number  to  be  multiplied,  and  adding  it;  thus,  40x30^= 
1200X10=1210. 

Example  1. — Reduce  1  square  mile,  2  acres,  2  roods,  to 
square  yards. 

640         4 
1  sq.  mi.     2  a.     2  r.   i 
640 

GlJa. 
4 


2570  r. 

40 

102800  sq.r.' 
301 


It  is  the  same  to 
take  ^  of  a  number, 
as  to  multiply  by  1 
and  theu  divide  by 
4;  thus,   4)102800 


gives 


25700 


3084000 
25700 
Ans.     3l09700sq.yd. 

2.  Reduce  10a.  5r.  3sq.  yd.,  to  square  feet. 

3.  Reduce  50a.  3r.  6sq.  ft.,  to  square  inches. 

4.  Reduce  25a.  14sq.  r.,  to  square  foet. 

5.  Reduce  45r.  8sq.  yd.,  to  square  inches. 

6.  In  3sq.  mi.,  how  many  square  feet? 

7.  In  2a.  2r.,  how  many  square  feet? 

8.  In  3sq.  yd.  2sq.  ft.,  how  mflny  s(}uare  inches? 


CIRCULAR  AND  ASTRONOMICAL  MEASURE. 

98.  r)0  Seconds,  60"  make  1  minute,  1' 
60  Minutes  make  1  degree,  1°. 
80  Degrees  make  1  sign,  .?. 
12  Signs,  or  360°  make  1  circle,  circ. 


f'ircul.'u-  an  1  :»< 
troiionuofil 


54 


REDUCTION. 

Reference  Table. 

s. 

o                        t 

60" 

1     =            1 

— 

3600 

1 

=      30    =        60 

:rr: 

108000 

12 

=    360    =     1800 

=. 

1296000 

1  = 

This  measure  is  used  to  calculate  latitude  and  longitude 
and  astronomical  distances, 
i^i'^i'^ions  «f  Circles  of  all  sizes  are  supposed  to  have  the  divisions  of 

m  'iinaJi.'''"^     360  equal  parts,  called  degrees;  these  degrees  are  divided 
into  60  equal  parts  called  minutes;  and  these  minutes, 
each,  into  60  seconds. 
Example  1. — Reduce  4  signs,  25°,  15',  to  minutes. 
30  60 

4s.        25°         15' 
30 

120 
25 

45° 
.60 


measure. 


8700 
15' 


Ans.     8715' 

2.  Reduce  5circ.  2s.  3°,  to  minutes. 

3.  Reduce  4circ.  15°  16',  to  seconds. 

4    In  Scire.  10°  14',  hew  many  s(^conds? 
,  5.  In  10°  15',  how  many  seconds? 
6.  In  6circ.  9s.  14°,  how  many  minutes? 


surveyor's  measure. 

sni veyoi's  99.        7  i^o^  Inches,  in.  make  1  link,  K. 

25         Links  maJce  1  rod,  perch  or  pole,  rd. 
4         Rods  make  1  chain,  ch. 
10         Chains  make  1  furlong,  yiwr. 
8         Furlongs  make  1  mile,  mi. 
Also, 
10,000         Square  links,  or  16  square  rods  make  1 
square  chain,  sq.  ch. 
10         Square  chains  make  1  acre,  a. 


EEDUCTION. 

Reference  Table, 

mi.     fur.        ch.          rd.            U.  in. 

1  =  1^ 

1  =      25  =  198 

•    1  =      4  =    100  =  792 

1  =  10  =    40  =  1000  =  7920 

1  =  8  =  80  =  320  =  8000  =  63360 

This  is  used  in  laying  out  railroads,  and  measuring 
the  boundaries  of  fields.  • 

The  surveyor's  chain  is  4  poles  or  66  feet  long;  it  is 
divided  into  100  links. 

Example  1. — Reduce  25  miles,  5  furlongs,  3  chains, 
10  rods,  to  links. 

8  10  4  • 

25  mi.     5  fur.     3  ch.     10  rd. 
8 

205  fur. 
10 

2053  eh. 
4 


65 


8222  rd. 
25 


41110 
16444 

Ans.  205550  li. 

2.  Reduce  8fur.  6ch.  9rd.,  to  rods! 

3.  Reduce  30mi.  3ch.  8rd.,  to  links. 

4.  Reduce  9ch.  3rd.,  to  linlcs. 

5.  Reduce  5mi.,  to  rods. 

6.  In  15  mi.  6fur.  3ch.  2  rd!,  how  many  rods? 

7.  In  7ch.  3rd.,  how  many  links?  * 

8.  In  30fur.  9ch.  5rd.,  how  many  rods? 

SOLID  OR  CUBIC  MEASURE. 

100.  1728  Cubic  inches,  cw.m.  make  1  cubic  foot,  c. /<.  ^^"^^g^;.  *"'^'* 
27  Cubic  feet  make  1  cubic  yard,  e.  yd. 


•'>6 '  REDUCTION. 


A   cord   meas- 
ure. 


40  Cubic  feet  make  1  ton  of  timber,  t. 
16  Cubic  feet  make  1  cord  foot,  eft. 
8  Cord  feet,  or  128  cubic  feet  make  1  cord, 


Reference  Table. 

cu.  yd.      cu.  ft.  cu,  in. 

1    =      1728 
1     =     27    =  '  46656 

This  is  used  to  measure  what  has  length,  breadth' and 
thickness. 

A  cord  of  wood  is  4  feet  wide,  4  feet  thick  and  8  feet 
long.  '    " 

A  cube  defined.  A  Cube  is  a  figure  of  six  equal  squares,  called  faces; 
the  sides  of  the  squares  are  called  edges.  The  face  on 
which  a  cube  stands  is  called  its  base.  If  the  edge  is  one 
yard,  it  will  contain  3x3=9  square  feet;  therefore,  9 
cubic  feet  can  be  placed  on  the  base;  and  hence  if  the 
figure «vere  1  foot  thick,  it  would,  contain  9  cubic  feet;  if 
•    •         it  were  2,  it  would  contain  twice  as  many;  if  3,  27  feet. 

Tofindttiecon-     The   Contents   of  a   cube    are    found   by   multiplying 

tents  of  a  cube,  together  the  length,  breadth  and  thickness. 

Theio.sainhew-      Round     timber     is    estimated    to    lose    one-fifth    by 

,ng  timber.       gquaring. 

Example  1. — Reduce  18  cubic  yards,  18  cubic  feet, 
]  5  cubic  inches,  to  inches. 


Ans. 


27 

1728 

18  cu.  yd. 

IScu.ft. 

15  cu.  in, 

27 

126 

18 

18 

324  cu.  ft. 

' 

1728 

2592 

648 

2268 

324 

659872  cu.  in. 

REDUCTION. 

2.  Reduce  Ic.  5cu.  yd.  20cu.  ft.,  to  inches. 

3.  Reduce  16cu.  yd.  15cu.  ft.,  to  feet. 

4.  Reduce  12cu.  yd.  lOcu.  ft.,  to  inches. 

5.  Reduce  5c.,  to  cubic  inches. 

0.  In  25  cords  of  wood,  how  m\ny  cord  feet ? 

7.  In  30  cords  of  wood,  how  many  cubic  feet  ? 

8.  In  120  feet  round  timber,  how  many  inches 


TIME  MEASURE.    , 

101.     GO  Seconds,  .sec.  make  1  minute,  m. 

60  Minutes  make  1  hour,  h.        '  "^'"'^^  "^^•'"'"•^. 

24  hours  make  1  day,  d. 

7  Days  make.  1  week,  wk. 

4  Weeks  make  1  lunar  month,  I.  m. 

12  Months  make  1  calendar  year,  c.  yr. 

13  Mouths,  I  day  and   6  hours  make  1  Julian 

year,  J.i/r. 


Reference  Table. 

I.  711. 

wh.           d.            h.              m.                sec. 

\=           CO 

1=        60=        3600 

1  =     24=     1440=      86400 

1     =     7  =  168=  10080=     604800 

1 

=  4    =  28  =  672=  40320=  2419200 

13, ^; 

,=52J^3±=365i=8766=525960=31557600 

J.r/it-. 


1  = 

Note. — As  the  length  of  the  year  is  365  days  and  6 
hours,  the  odd  hours  in  4  years  make  1  day,  which  is 
added  to  every  fourth  year,  in  the  month  of  February.  '^<'^r  J*''""- 
The  year,  thus  increased,  is  called.  Leap  Year. 

Years  exactly  divisible  by  4,  as  1860,  1864,  1868,  are 
leap  years. 

The  following  verse  memorized,  is  of  use  to  recall  the 
number  of  days  in  each  month : 

Thirty  days  hath  September,  TheriavKof  tiie 

April,  June  and  November;  * 

All  the  rest  have  thirty-one 

Except  the  second  month  alone, 

And  that  has  eight  and  twenty,  clear, 

But  nine  and  twenty  each  Leap  Year. 


moath. 


5& 


EEDUCTION. 


Example  1. — Reduce, 5  years,  9  months,  6  days,  12 
hours,  15  minutes,  to  minutes. 

12  7  24  60 

5  yr.     9  mo.     6  da.     12  hr.     15  m. 
12 

In  this   sum  let  it 


2791440 
15 


69  mo-. 
4 

be  noted  that  no  weeks 
are  named. 

276  wk. 

7 

1938  da. 
•   24 

7752 
3876 
12 

46524  hr. 
60 

Ans.     2791455  m. 


102.  2.  Reduce  IJ.  yr.  to  minuted. 

3.  Reduce  5mo.  5da.  16hr.,  to  minutes. 

4.  In  25yr.  2wk.  5da.,  how  many  hours  ? 
6.  In  Ida.  18hr.  20ra.,  how  many  seconds  ? 

6.  In  15hr.  35m.  40sec.,  how  many  seconds? 

7.  Reduce  Ic.  yr.  to  minutes. 

8.  Reduce  3mo.  5wk.  5da.,  to  hours. 

9.  Reduce  2J.  yr.  to  minutes. 
10,  Reduce  Se.  yr.  to  hours. 


REDUCyON. 


id 


MISCELLANEOUS  TABLES. 

103.      12  Units,  or  things,  make  1  dozen. 

12  Dozen  make  1  gross. 

12  Gross,  ox  144r  dozen,  make  1  great  gross. 

20  Units,  or  things,  make  1  score. 

196  Pounds  make  1  barrel  of  flour. 

100  Pounds  make  1  quintal  of  fish. 

200  Pounds  make  1  barrel  of  pork. 

18  Inches  make  1  cubit. 

14  Pounds  of  iron  or  lead  make  1  stone. 

*  21-5  Stones  make  1  pig. 

8  Pigs  make  1  fother. 

24  Sheets  of  paper  make  1  quire. 

20  Quires  make  1  ream. 


i04. 


FOREIGN  COINS- 


COUNTRY. 

'  •     GOLD  COINS. 

SILVER   COINS. 

I>EN0MI.VATl6x. 

1    VAlDf,. 

BENOMINAWON'. 

1  v.\h;e. 

Austria,    . 

Ducat,    . 

•Z  28  0 

Seudo.    . 

3    Ct«.    IB, 

1  01  5 

Belgium, 
Bolivia,     . 

25  Francs, 

4  72  0 

5  Francs,   . 

9t>  8 

Doubloon,     . 

15  .58  0 

Dollar.   . 

1  05  4 

Brazil,  . 

:i(J.ooo  Kois, 

10  90  .5 

2000  Reis.  . 

1  01  ;j 

€hi!i, 

\0  Pesos, 

0  15  .3 

New  Dollar,  . 

97  0 

Denmark,     . 

10  Thaler.  . 

7  flO  0 

2  Kigsdaler,      . 
Shilluig.  new. 

1  09  4 

England, 

Sovereigil,  now,  . 

4  t!fl  3 

22  7 

England, 

Sovereign,  average. 

4  84  8 

Shilling,  average. 

22  2 

France,     . 

20  Francs,  average, 

3  84  o 

5  Francs,  average, 

90  8 

.Germany,  nortli. 

10  ThsJer, . 

7  00  0 

Thaler, 

71  7 

Germaay,  south, 

Dueat,    . 

2  28  3 

Guilder  or  Florin, 

41  2 

Mexico. 

Doubloon,  average, 

15  53  4 

Dollar,  avcr.ige, 

1  04  9 

Netherlands,  . 

10  Guilders,  . 

3  99  0 

2i4(jNUders, 

1  02  3 

New  Granuda, 

10  Peso.s.  new.  . 

9  0"  5 

Dollar,  1857,      . 

96  8 

Peru, 

Doubloon,  old,     . 

35  56  0 

I>ollar,  1855, . 

93  r, 

Portngal, 

Crown, 

5  81  3 

Crown, 

1  16  (4 

Rome, 

2y,  Sendi,  new,    . 

2  60.0 

.Scudo,   . 

1  04  7 

Russia, 

5  Roubles, 

3  97  6 

Rouble.      . 

78  4 

Spain, 

100  Heals,      . 

4  90  3 

Pistareen,  new,    . 

20  1 

Sweden, 

Dueat, 

2  20  7 

Rix  dollar. 

1  10  1 

Turkey,    . 

100  Pi.istres, 

4  37  -J 

20  Piastres,,  . 

86  5 

TuHcnny, 

Seqnin. 

2  .30  n 

Florin, 

27  4 

iVo^r. — The  above  values  are  computed  at  the  Mint  rate 
of  $18.60  per  ounce  standard  (9-10  fine)  for  gold,  and 
^1.21  per  ounce  standard  for  silver. 

Note. — The  English  pound,  or  pound  sterling,  is  valued 
at  e4.44c.  4m.  (Art.  285).  The  French  franc  is  valued 
at  18. i  cents  (Art.  286). 


BEDUjCTION. 


»  RULES  FOR  MEASURING  CRIBS,  HOGSHEADS,  ETC.      V"^ 

To  find  the  number  of  cubic  feet  iu  any  square  crib  or 
box,  multiply  the  length  by  the  breadth  (in  I'eet)  for  the 
number  of  square  feet  on  the  floor,  and  this  product  by 
the  depth,  for  the  required  number  of  cubic  feet  in  the 
box  or  room.  Thus  if  a  room  be  12  feet  long  by  6  wide, 
it  contains  12x6=72  square  feet  on  the  floor,  and  if  5 
feet  deep,  it  contains  72x5=360  cubic  feet. 

To  find  the  number  of  bushels:  A  cubic  foot  contains 
1728  cubic  inches — and  a  bushel  about  2160  (accurately, 
2150.42)  inches.  A  cubic  foot  is  therefore  1728-2160= 
4-5  or  8-10  of  a  bushel.  A  wine  gallon  contains  281 
cubic  inches.  A  cubic  foot  therefore  contains  about  7  1-2 
and  a  bushel  about  9  1-3  wine  gallons. 

Corn  is  usually  put  up  on  the  cob  or  in  the  shuck, 
while  it  is  sold  by  the  bushel  or  barrel  of  shelled  corn. 
The  proportion  of  shelled  corn  to  corn  on  the  cob  is  nearly 
uniform,  but  compared  with  corn  in  the  shuck'  it  varies 
considerably — depending  on — 1,  the  size  of  the  ears — 2, 
the  Avay  it  is  shucked,  and — 3,  the  way  it  is  packed  or 
troddcu  in.  One  bushel  of  shelled  corn  is  equal  to  two 
bushels  of  corn  on  the  cob,  to  about  three  bushels  of  corn 
in  slip  shuck  (say  21  to  3i),  and  to  about  4  of  corn  in  full 
shuck  (say  4  to  4^). 

If  a  crii)  of  corn  on  the  cob  is  12  feet  long,  10  wide,  and 
8  deep,  it  will  hold  as  follows: 
12      Lensrth  in  feet. 
10      Widlh. 

120      Square  feet  on  floor. 
8      Depth. 

960      Cubic  feet, 

8      =8-10  Multiplier  for  bushels. 


7680      (The  right  hand  figure    cut  ofl)  number  of 
bushels  of  corn  on  the  cob — 768. 


2—7680 


384  Number  of  bushels  of  shelled  corn. 

S — 768  Bushuls — if  in  slip  shuck. 

256  Bushels  of  shelled  corn. 

4 — 768  Bushels — if  in  full  or  whole  shuck. 


REDUCTION. 

192      Bushels  of  shelled  corn. 
5 — 38-4      Bushels  of  shelled  corn. 

7G  4-5  Barrels  of  shelled  corn. 
\Nbte. — If  the  corn  be  not  level,  it  must  be  made  so  or 
averaged. 

A  concise  rule  for  finding  the"  contents,  in  shelled  corn, 
of  a  crib  of  corn  put  up  in  the  cob. 

Multiply  together  the  length,  breadth  and  average 
depth,  expressed  in  feet.  Multiply  this  product  by  4, 
and  cut  off  one  figure  from  the  right,  for  the  answer  in 
bushels  of  shelled  corn. 

EXAMPLE. 

In  a  crib  15  feet  long,  12  feet  wide,  filled  9  feet  deep 
with  corn  on  the  cob,  how  many  bushels  of  shelled  corn  ? 
15     Length  in  feet. 
12     Width. 

180     Square  feet  on  floor. 
9     Depth. 


1620     Cubic  feet. 

4     Multiplied  for  bushels. 


648,0     (One  decimal  cut  off)  684  bushels  shelled  corn. 

If  the  corn  be  in  slip  shuck,  multiply  the  cubic  feet  by 
o,  and  if  in  full  shuck,  by  2,  and  cut  off  one  figure  as 
decimal  for  the  answer  in  bushels  of  shelled  corn. 

A  concise  rule  for  reducing  corn  on  the  cob,  to  barrel?? 
of  shelled  corn  : 

Take  8  per  cent,  of  the  product  of  length,  width  luid 
depth,  expressed  in  feet. 

EXAMPLE. 

In  a  crib  of  corn  on  the  cob  20  feet  long,  10  wide,  and 
9  deep,  how  many  barrels  of  .shelled  corn  ? 
20X10=200—200X9=1800 

8  per  cent. 

14400  cut  off  2  decima]s=^144 
bbls.  i 

For  all  grain,  wheat,  shelled  corn,  etc.,  which  are  sold 
as  they  stand,  multiply  together  the  length  and  bre;ulth 
and  depth  in  feet  for  the  number  of  cubic  feet;  multiply 


REDUCTION. 

this  by  8  and   cut  off  one   decimal  for  the  answer   in 
bushels. 

EXAMPLE. 

A  box  of  wheat  is  12  feet  long,  4  wide  and  5  deep: 
how  many  bushels  does  it  contain  ? 
12x4=48— 48  x5=:=-240 


192,0— 192  bushels. 

RULE  FOR  PEAS  IN  THE  SHELL. 

Multiply  together  the  length  and  breadth  and  depth  in 
feet  for  the  number  of  cubic  feet;  divide  this  product  by 
20  for  the  number  of  bushels  of  shelled  peas. 

EXAMPLE. 

In  a  room,  of  unahelled  peas,  20  feet  long,   15  wide, 
and  averaging  6  feet  deep,  how  many  bushels  of  shelled 
peas  ? 
20X  15=300.— 300X 6=1800.— 1800— 20=90  bushels. 

To  find  the  number  of  bushels  in  a  hogshead,  barrel,  or 
other  vessel  of  a  circular  base,  and  approximating  a 
cylinder  in  form,  measure  the  inside  diameter  one-third  of 
the  jfi^ay  down  from  the  top,  and  the  depth,  in  inches 

RULE. 

Multiply  the  diameter  in  inches  by  itself,  and  the  pro- 
duct by  the  deptli.  Then  multiply  by  86  jl,  and  cut  of  5 
decimals  for  the  answer  in  bushels. 

EXAMPLE. 

In  a  hogshead  whose  depth  is  40  inches,  and  the 
diameter  (one-third  from  the  top)  30  inches,  how  many 
bushels  ? 

30     diameter  in  inches. 
30 

900 

40  depth. 


36000 
36} 

216000 
108000 
18000 


1314000  (5  decimals)— 13  bushels,  14-000. 


REDUCTION.  •  ^^ 

To  find  the  nutnber  of  bushels  in  a  potato  bank,  piled 
in  the  form  of  a  cone : 

RULK. 

Multiply  the  diameter  at  the  base  by  itself,  and  the 
product  by  the  height  in  feet.  Then  multiply  by  21  and 
cut  oif  2  figures  for  decimals  for  the  answer  in  bushels. 

EXAMPLE. 

In  a  potato  bank,  the  diameter  being  6  feot  at  the  base, 
and  the  height  5  feet,  how  many  bushels  ? 
6x6=36—36x5=180 
21 

180 
860  .         ^ 

3,780=37  8-10  bushels. 
If  the  potatoes  do  not  come  to  a  poiiit  at  the  top,  but 
round   considerably;  then   divide  the   180  by  4  for   the 
answer—say  180— -4::i=45  bushels. 

TO    MEASURE  BY  A  MEASURlNCi  ROD. 

Cut  a  rod  exactly  51 1  inches  long,  and  measure  it  oflF 
into  4  equal  parts.  Each  part  \\^11  be  a  line  *  bushel.  A 
box  just  as  long,  wide  and  deep  as  this  would  contain 
exactly  one  bushel.  Subdivide  each  line  bushel  into  ten 
equal  parts  calling  them  tenths. 

When  the  dimensions  are  found  with  this  rod,  the 
product  of  length,  breadth  and  depth  is  the  answer  in 
bushels.  ♦ 

EXAMPLE. 
A  crib  is  10  line  bushels  long,  8  wide,  and  5  4-10  (or 
5-4)  deep  :  how  many  bushels  does  it  contain  ? 
10     line  bushels  long. 
8     wide. 

80 
54 


320 

400 


432,0     number  of  busbels— 432. 


*  12  906-1000  inches. 


64  REDUPTION. 

If  tlie  crib  is  full  of  corn  on  the  cob,  divide  by  2  to 
reduce  it  to  shelled  corn,  and  so  in  other  cases. 

Example  1. — How  much  clear  corn  in  a  bin  5  feet 
high,  8  wide  and  16  long,  the  corn  being  in  full  shuck? 

2.  How  much  clear  corn  in  a  bin  12  feet  wide,  10  high 
and  20  long,  the  ears  being  in  slip  shuck  ? 

o.  What  is  the  quantity  of  corn  in  a  bin  61  feet  wide, 
8i  feet  high  and  101  feet  long,  the  ears  being  without 
shuck? 

4.  How  many  bushels  of  peas  in  a  room  20  feet  wide, 
30  long  and  42  high? 

5.  How  many  bushels  of  potatoes  in  a  bank,  the 
diameter  at  the  base  being  8  feet  and  the  height  7  feet? 

6.  A  erjb  is  9  line  bushels  long,  7  wide  and  3  3-10  (or 
3,  3)  deep:  how  many  bushels? 

105,  To  change  numbers  from  a  lower  to  a  higher 
a  '  iowtn^  to""ttname,  without  change  of  value,  we  employ  division.  This 
Higher  uama.     jg  jj  giinpje  reverse  of  the  process,  used  in  reducing  from  a 

higher  to  a  lower  denomination. 

Example  1.  Keducc  10354  farthings  to  £.* 

OPERATION.  I  EXPLANATION. 

<.f  4)10354  I       We  first  divide  by 

4,  since  there  can  be 

12)2588  2qr.  '         only     one- fourth     as 

• •  many    pence    as    far- 

20)215  8d..  things.     By  this  divi- 

sion  are  found  2588d, 

Ans.  £10  15s.  8d.  2qr.       and    a    remainder    of 

2qr.  To  reduce  these  pence  to  shillings,  we  divide  by  12, 
since  there  can  be  only'  one-twelfth  as  many  shillings  as 
pence,  and  we  get  215s.  and  also  have  a  remainder  of  8d. 
To  reduce  these  shillings  to  pounds,  we  divide  by  20,  and 
find  for  result,  £10  and  a  remainder  of  15s.  Placing  by 
the  side  of  this  figure,  £10,  the  several  remainders  in 
their  proper  order,  we  find  that  10354  farthings  are,  when 
reduced,  £10  15s.  8d.  2qr. 

106.  In  a  similar  manner,  like  examples  are  to  be  per- 
formed according  to  the  following  directions: 

Divide  the  sum  directed  to  be  reduced  by  the  number 
tiom  a  lower  to  of  its  denomination  that  makes  it  higher;  divide  that 
»  higin-r  nai!ie.  ^^g^jj,  ^^,  ^^^  number  of  its  denomination,  and  so  on.    The 

final  quotient  and  the  several  remainders  are  what  were  to 

be  found. 


otlanatiou 
-fk. 


How  to  i-fdueo 


*  Ex.  1.,.  Art.  88,  shows  the  reverse  process. 


pren 


REDUCTION.  b-' 

107.  It  will  be  seen  by  the  coiiiparison  of  Ex.  1,  Art.  The  two  kin.i* 
88,  with  Ex.  1,  Art.  99,  that  the  two  kinds  of  reduction  X'*"'"'^ 
prove  each  other. 

Example  1. — Reduce  1365  inches  to  rods. 

OPERATION. 

12)1865  Note. — In  this  ex- 

ample  it  will  be  seen 

3)113  ft.  9  in.  that  the  division  by 

51  is  easily  performed 

5])37  yds.  2  ft.                   I  by      doubling      that 

—  I   number,      and      the 
11)74  I  dividend,  37,  so  that 

—  I  they     stand     11)74. 

Ans.      6  rods,  4  yd.  2  ft.  0  ill. I  Such    change    never  Diyi,iing ^.y  r,v<, 
affects  the  value  of  a  term,  while  it  relieves  from  the  o^;^^''.'^^''/ •'-''■ 
embarrassment  of  fractional  division.     A  similar  method 
is  to  be  observed  in  dividing  by  any  number  connected 
with  \,  h,  etc.,  only   changing  the  divisor  and  dividend 
into  fourths,  thirds,  etc.,  as  the  case  requires. 

2.  Reduce  455  pints,  Dry  measure,  to  higher  denomi- 
nations. 

3.  Ucduce  29795  cubic  inches  to  feet  and  yards. 

4.  lleduce  177564  farthings  to  £. 

5.  lleduce  65432  shillings  to  £. 

6.  lleduce  59678  pence  to  shillings. 

7.  lleduce  965480  pence  to  £. 

8.  Jlcduce  3764354  pounds  to  tons. 

9.  Reduce    545509    grains,  Apothecaries'  weight,  to 
pounds. 

10.  In  2500  nails,  how  many  yards? 

11.  Reduce  5665  rods  to  miles. 

12.  Reduce  3567  links  to  miles. 

13.  In  9657840  cu.  in.,  how  many  cords? 

14.  Reduce  985067  sq.  r.  to  sq.  miles. 

15.  What  number  of  circles  in '1296000  seconds? 

16.  How  many  gallons  in  835  gills? 

17.  In  896574  seconds,  how  many  calendar  months? 

18.  In  765325  ounces,  how  many  tons? 

19.  In  57850  links,  how  many  miles? 

6 


tiG  REDUCTION. 


MISCELLANEOUS  examples; 

08.  Example  1. — Reduce  £,75  15s.  dd.,  to  pence. 
.  2.  Reduce  £5  6s.  7d.  3qr.,  to  farthings. 

3.  Reduce  6078095  farthings  to  £. 

4.  Reduce  695432  pencp  to  £. 

.'■.  Reduce  6bush.  2p.  6qt.  Ipt.,  to  pints.  ,   ^ 

«■.  Reduce  50t.  15cwt.  2qr.  161b.  Tioz.  8dr.,  to  drachiv.^. 

7.  Reduce  3c.  yd.  20c.  ft.  1435c.  in.,  to  cubic  ijiche?^. 

;>.  Reduce  6mi.  5fur.  6ch.  2rd.  161i,,  to  links. 

9.  Rcdiiec  141b.  95-  55-  29-  16gr.,  to  grains. 

10.  Reduce  651b.  9o^.  15pwt.  ISgr.,  to  gTaii;8. 

11.  Reduce  5yd.  2qr.  3na.,  to  nails. 

12.  Reduce  5^89 65  links  to  miles. 

13.  Reduce  694205c.  in.  to  cubic  yards, 

14.  Reduce  787650na.  to  yards. 

15.  Reduce  3yd.  2ft.  6in.  2b.  c,  to  barley  corn.s. 

16.  Reduce  4m.  4fur.  25rd.,  to  rods. 

17.  Reduce  26sq.  m.  30a.  2rd.  33sq.  rd.,  to  square  rods. 

18.  Reduce  9gal.  3qt.  2pt.  3gi.,  to  gill^. 

19.  Reduce  6378  gills  to  gallons. 

20.  Reduce  12circ.  e.s.  15°  45'  10",  to  seconds. 

21.  Reduce  5mo.  3wk.  16h.  27sec.,  to  seconds. 

22.  Reduce  3m.  6ch.  ^rd.,  to  links. 

23.  ReducQ  3t.  181b.,  to  ounces. 

24.  Li  75a.  Gx-  5sq.  rd.,  how  piany  square  inches  ? 

25.  In  30mo.  3w.  6da.  i5h.,  how  many  minutes? 


AMERICAN    MONEY.  67 

* 

AMERICAN  MONEY. 

SECTION    III. 

W}>  Arr:ran3fanr^/h   tlv.^  currencv  of  tlio  Soutlieru  g^Sg^*;!,"'" 
iifcclcracy.  ■        ,    Statesi  ■ 

no.    It  ^:-^    '■-•'■ -    ->.o-^^.   ■V.IImv-^.  K^.ii/isions. 

lies,  cents  and  mills 

ni.  The  .rroM  ■"■  .u".....i.v 

_;lc  and  dollar. 

112i   The  silver  cniii   is   me  (.ioiiar,  iiaii-*;(.)i;aV;(  f|i:;v. 

iiav.  dime,  half-dime,  aud  threo-cerit  piece. 

il3.  The  nickel  is    '      >  numeicially 'ciilled  tiic,'-'"*!'  • '^'   ■ 

,  ,.  .  J;   ■  ly  ii  copper 

, 'pev.  '  -  .      ^  ■  ^   ■  co:r. 

AME*IICAN   MONET  TABLE. 

I.14,  10  mills  luake  1  cent,  marked  cf. 

10  cents  make  1  dime,  marked  d: 
10  dimes  make  1  dollar,  marked  >'■< 
10  dollars  make  1  eagle,  marked  U. 


K'j^ih.       Jh. 


1       = 
<f 

115,  As  this  money  comes  under  the  directions  for  per-  _ 
forming  addition,  subtraction,  multiplication  and  division, /oiij,.,       " 
the  general  subject  of  this  currency  is  referred  to  them. 

110.  A  point  is  used  to  separate  dollars  from  cents ;  Tho.  {)<„nt  dis- 
thus  ^5.66  is  read  5  dollars  and  sixty-six  cents:  ^Mthout  the  ^  aS'ii'^c^ntl  "" 
point,  S5(i6  would  be  five  hundred  and  sixty-?ix  dollar?. 

117,  When  three  figures  are  gt  the  right  of  the  point,  The  fig-iros  t-x 
the  two  first  are  c;ents,  and  thci  third  figure  is  mills;  thus,  P''^^^"''^«  °''"'^'' 
S5.666  expresses  five  dollars,  sixty-six  cents,  six  mills.    If' 
a  comma  had  been  where  the  point  stands,  it  would  be  read 
<h'o  thousand  six  hundred  and  sixty-six  dolla,rs. 


Reference  Table. 

/)rin-'n.          Cetifs. 

Mi'h:. 

1     = 

'10 

1     =       10     = 

100 

1 

=         10     =     100     = 

1000 

0 

=     1000    =     100     = 

10,000 

68  AMERICAN   MONEY. 

How  parts  of  a     118.  Parts  of  a  dollar  are  often  expressed  in  numbersy 
J?i?8ed.'^  ^""^   l^DOWQ  as  fractions,  thus  : 

50     cents,  or  a  half-dollar,  is  written    -    .     -         -     i 
83J  cents,  or  a  third-dollar,  is  written      -         -         i 
25     cts.jOrone-fonrthor  one-quarter  dollar,  is  written  J 
20     cents,  or  one-fifth  dollar,  is  written       -         -     i. 
12i  cents,  or  one-eighth  dollar,  is.  wri.tten         -         i 
10     cents,  or  one-tenth  d»llar,  is  written      -         -     ^o 
6}  cents,  or  one-sixteenth  dollar,  is  written     -        ^^ 
5     cents,  or  one-twentieth  dollar,  is  written        -    ^o 
5     mills,  i  of  a  cent. 
Cents  chaneed      119.  Cents    become  mills   by  the  annexation  of   one 
u>*cents  and" cipher;  dollars  become  cents  by  the  annexation  of  two 
■Bills.  ciphers,  and  mills  by  three;  and  eagles  become  dollars  by 

changel'to  doi-  ^^^  annexation  of  one  cipher ;  thus,  60  cenis  are  600  mills ; 
•*r8.  65  dollars  are  6500  cents,  65,000  mills  ;  75  eagles  are  750 

Examples  of      ,   „  '        '  )  a 

ohaiige.  dollars. 

Example  1. — Write  45  dollars,  46  cents,  6  mills,  in 
mills. 

OPERATION. 

45  dollars=4500    cents. 
Add  to  these     46    cents. 


4546   cents. 

Annex  one  cipher  to  change  to  mills,    45460  mills. 

Add  6  mills. 


Anst.     45,466  mills. 
2.  In  35  dollars,  63  cents,  5  mills,  how  many  mills  ? 
^.  In  865,  how  many  cents  ? 
4    In  $550,  how  many  cents  ?  how  nsany  mills  ? 

5.  In  S5.60,  how  many  mills  ? 

6.  In  82.50,  how  many  cents  ? 

7.  In  $100,  how  many  cents  ? 

8.  In  5  E.  and  §6.40,  how  many  cents  ? 
Mills  changed       120i  To  change  mills  to  cents,  the  right  hand  figure 
joints;  todoi-  ^^^^  ^^  ^^^  ^g..   ^^^  ^^jjj^  ^^  (jollars,  the  three  right  hand 

figures. 
Cents  changed      121*  To  change  cents  to  dollars,  the  two  right  hand 
to  dollars.        figures  must  be  cut  off. 

Remark. — When  dollars  are  multiplied  by  dollars,  the 
answer  is  in  dollars ;  when  by  cents,  the  answer  is  in  cents ; 
and  when  cents  are  multiplied  by  cents,  the  answer  is  in 
mills. 


AMERICAN    MONEY.  69 

9.  Change  36445  mills  to  dollars,  cents  and  mills. 

Ans.  36,45,0,  or  e?6.45c.  5m. 

10.  Change  6954320  mills  to  dollars,  cents  and  mills. 

11.  Change  78654461  mills  to  dollars  and  cents. 

12.  Change  5567905  mills  to  dollars  and  cents. 

13.  Change  3890076  mills  to  dollars,  cents  and  mills. 

14.  Change  3650123  mills  to  dollars  and  cents. 

15-  Change  984060  mills  to  dollars,  cents  and  mills. 
122«  The  mill  is  simply  an  imaginary  coin,  and  in  com-  The  mill  an  im- 
mcrcial   transactions  hardly  known;   thus,  in   the  sale  of *g'"'"'>' <^<''°- 
articles,  amounting  severally  to  62  J 

431 
1.3U 


Which  added,  are  $2,371 

The  trader  does  not  express  the  J,  I  and  J  in  mills,  but 
adds  them  as  f'ractions,  and  writes  the  same  as  above,  not 
as  the  amount  equally  is,  $2. 37c.  5m. 

Remark. — The  pupil  must  be  particular  in  putting  thepojnt^nof  U)'b« 
separation  point  between  the  dollars  and  cents;  as  also, J\<'gi«'^*«d;»'»0' 
whert  adding,  to  place  cents  under  cents,  and  dollars  under  ment  of  eimiiar 
dollars.  _  ^■*"^«*- 

123*  When  one  figure  only  expresses  the  cent,  a  cipher  How  to  write  m 
is  to  be  placed  at  its  left;  thus  to  write  four  dollars  and  ^^^^^  ^^*^'* ' 
six  cents,  we  d^  not  write  $4.6,  but  $4.06. 

124i  Iq  t.he  following  examples,  in  the  Addition  o/A«idition  of 
American  Monet/,  the  answers  can,  be  found  by  the  table,  ^6^^'.*^*°  ^^^' 
or  in  the  commercial  form  of  adding  fractions. 

Example  1. — John  bought  6  pair  socks  for  $1.25,  a 
vest  for  $2.25,  a  coat  for  $9  37^,  6  handkerchiefs  for 
$1.62  J,  and  a  cravat  for  75e  :  what  was  the  cost  ? 

OPERATION. 


Socks,               $1.25 

= 

$1.25      or    $1.25 

Vest,         .        •  2.25 

= 

2.25 

2.25 

Coat,                    9.371 

= 

9.375 

9.371 

Handkerchiefs,    1.621 

= 

1.625 

1.621 

Cravat,                   .75 

= 

.75 

.75 

m 


Ans.         $15,250  $15.25 

2.  Purchased  1  box  of  candles,  for  $7.50;  1  box  raisins, 
$3,371;  1  keg  of  buckwheat,  $2,621;  1  barrel  of  flour, 
$9,871  :  what  was  the  amount  of  the  purchase  ? 

3.  Bought  1  bag  of  coffee  for  $15.621 ;  5  sacks  of  salt, 
$4,371;  1  barrel  of  molasses,  $14.45,  and  1  box  of  starch, 
$3.371 :  what  was  amount  of  bill  ? 


70  AMERICAN    MONEY. 

4.  If  you  owe  to  A  $437.50^  .to  B  665  ;^to  G.|5.3YJ  j  t. 
i)  02 J  :  what  is  the  whole  amount?  , 

5.  If  your  father's  State  tax  is  819.37^;  liis  town  tax 
S25.12i ;  his  poor  tax  |G.62i,  and  his  hridge  tax  $5.87 -i  : 
what  is  the'  sum  of  the  whole  ? 

^  G.  Add  $125,  $t)5.37|,  $60.62.1,  and  ei235.87Hogether; 

Subtraction  of       125,  The    fSuhtractton  of  American  money  is  suhstar;- 

Aineriftau  cur-    j.-   ^^     j.-i     i     i-     •        ^  i 

reroy.  tially  that  or  .simple  numbers. 

Example  1. — If  my  income  is  $2500  a  year,  and  ni\ 

pv,,f>n(i;inves  are  62437.50:  what  is  the  surplus  ? 

:  ;:hatioiJ.  explanation. 

2500.00  I       Having  for  convenience  put  the  two 

2487.50  I  ciphers  in  the  place  of  cents  in  tlu 

-- — : I  minuend,  we  say  0  from  0,  nothing, 

$62.50  i  which  set  as  a  cipher  in  the  units- 

phuc  of  cents,  we  then  say  5  from  \eu  (Art.' 40,  remark) 

5 ;  placin<^  this  in  the  tens-place  of  cents,  we  carry  the  1 

]jorrowed  to  7,  and  then  proceed  with  the  subtraction  as  in 

the  ruje  for  simple  numbers. 

2.  A  man  buys  a  horse  for  387.50,  what  change  is  h%  to 
•    receive  from  a  hundred  dollar  bill  handed  the  seller  ? 

3.  If  yon  pay  for  a  cari-iage  $450,  and  for  a  pair  of 
lioi'.ses  ?>337.7''^  ^'""v  v-"'--i>  m,.,-,.  ,1,,,.c:  flw.  '-arriage  cost  than 
the  htn-ses?  , 

4.  AVhat   is     lIh.-    u!ii'.n.-in.c    ueiwrL-u    ?775.37l',    and 

!^5e2.r2i?  • 

5.  Wh;;t  would  8595  deducted  from  $1000.50  leave  ? 
G.  How  much  more  is  62000.60  than  8999.99  ? 

7.  Deduct  $735.39  from  $862.21,  and  state  the  remain- 
der ? 

8.  What  is  the  differcnoe  between  $59.69,  and  $96.95  ? 

The  miiitipiiciv      126.  The   Multiplication  of  American  'money  is  similar 
!'«"  cfr-"lf V.   i'l  process  to  that  of  -simple  numbers. 

Example!. — What  will  28  pieces  of  cotton  bagging 
cost  at  $15.50  a  piece? 
operation.' 

15.50    I      Kemakk. — When  the  multiplicand, 
1^  t?K^  nmuipif-  28    i  as  in  this  example,  has  cents,  the  two 

>i.  I  right  hand  figures  in  the  result  must 

12400    i  be  separated  by  the  point  for  cents. 

3100   ; 


Ans.     $434.00 


AMERICAN    MONEY. 

2.  What  is  the  cost  of  40  barrels  of  liouv  at  $C)i  a 
barrel ? 

'  -PEKATiON.  I       Remark. — Here  4CX  J=forty  halve.s=20 
40         i  wholes;  or  it  could  be  said  §6 ■  =86.50 
G^       !  40 


240         I  .  $260.00 

20         I 

S260         I 

2    What  is  the  cost  of  a  firkin  of  butter,  containing^ 

libs.,  at  25 i  per*  lb? 

.'].  What  will  350  bushels  rough  rice  cost  at  .87  i  per 
bushel '( 

Eemark. — If  this  operation  is  performed  by  writing  When  tJio  mui- 
.875,  nameil,  when  so  written,  decimals,  for  the  multiplier, *^?-,!'^or*'dep;-"^ 
which  is  87  cents  5  mills,  three  figures  on  the  rigV  Uaua  J^^''-;-  <*f5"«!  f" 
side,  in  the  result,  are  to  be  marked  off,  the  first  on  the  '  •    ' 
(Extreme  right,  by  a  comma,  fur  mills ;  the  next  itwo  by  a 

'int  for  cents.     Our  preference  is  for  the  other  form,  ais 
i  'ing  in  common  use,  and  practically  best.  '    .  , 

4.  I  purchased  a  flock  of  sheep,  numbering  225,at'^2i 
t  acli :  what  did  the  whole  cost  'I 

5.  What  will  66  bushels  of  oats  cost  at  .33  J  per  bushel  ? 

6.  What  must  I  pay  for  52  barrels  of  potatoes,  at  §3] 
]ier  barrel'? 

7.  What  will  oSO'acres  of  land,  cost,  at  f;151  per  acre? 

8.  How  much  has  to  be  paid  for  20  railroad  share?, 
valued  at  $95,875  each? 

127.  To  find  the  cost  of  articles  soldV  the  100  or  1000,  T»  smi  tho  eosi, 
■il'tev  multiplying  th'e  quantity  by  the  pricey  we  cut  off  two  pp^^ioo'^  ^^^ -fi^oi. 
figures  on  the  right  hand  of  the  product,  if  the  price  be 
by  the  XOO;  and  three,  if  by  the   1000;  the  remaining 
figures  r%jresent  the  answer,  in  the  same  denomination,  as 
the  price.  ,         '  '  ■  ' 

9.  What  will  5750  bricks  cost  at  $10  .per  thousand  ? 

5750 
10 


Am.  $57,500 
or 
$57.50  c.  0  mills. 


*  Per,  the  Latin  particle,  signifying  for. 


71:  AMERICAN    MONEY. 

10.  Bought  a  raft  of  boards,  containing  3345  feet,  at  $12 
per  thousand ;  what  did  the  same  anaount  to  ? 

11.  What  is  the  value  of  3475  feet  of  timber  at  $2  per 
hundred  ? 

12.  What  must  be  paid  for  450  feet  of  boards  at  $8  per 
thousand  ? 

Articles  sold  by      '^^  ^^^  ^^^  worth  of  articles  sold  by  the  ton  :  having 

♦he  toi^.         '  multiplied   by   the   given   numbers,  we   strike  off  three 

figures    from   the   right  of  the  product,   and   divide  the 

^^^'^-.'o!!"''"""  '^^  remainder  by  2  for  the  answer.     This  answer  will  be  in 

fho  13tll  SUnii  ,  ,      •'  .  .  1  •  n  mi 

the  same  denomination  as  the  price  oi  a  ton.  ihe  reason 
of  this  division  is,  because  the  ton  consists  of  20001bs., 
and  the  example  proposes  a  number  less  than  that. 

13.  What  cost  1637  weight  of  blades,  at  $10.50  the 

ton  ?  OPERATION. 

1637 
1050 


81850 
16370 

2)1718,850 

A71S.    ^8.59 

14.  What  will  be  the  cost  of  26761bs.  of  plaster,  at 
$2.65  per  to»? 

15.  What  will  9501bs.  of  hay  cost,  at  $12.50  per  ton? 

16.  What  will  be  the  freight  of  56781bs.  of  iron,  at  $9 
per  ton  ? 

17.  What  will  be  ihe  cost,  by  railroad,  from  Charleston 
to  Memphis,  on  an  invoice. of  merchandize,  weighing  8560 
tons,  at  S7  per  ton  ? 

>)ivi5ion  of  128.  The  Division  of  American  money  is  to  be  pcr- 

Amcricau  cur-   fo,.„,g^^  ^^  j^  ^^jj^plg  Q^nibers.  _  i 

Dow  to  divide      129.  When  the  sum  to  be  divided  consists  of'dollars, 
tioUais.  annex  two  ciphers  at  the  right,  in  the  place  of  cents,  plac- 

ing, always,  the  separation  point  between  the  dollars  and 
cents.     The  answer  will  be  in  dollars  and  cents. 
v,hat  is  .ione       130.  Should  there  be  a  remainder,  it  is  to  be  expressed, 
ivhen  there  is  a  fractionally,  as  in   the   following  example;    or  if  it   be 
desirable  to  pursue  the  i.nquiry  further,  by  annexing  a 
cipher  to  the  dividend,  the  next  division  will  give  mills, 
and  so  on. 
riie  foi-mer,  or     REMARK. — The  first  way  IS  the  preferable  one,  for  the 
•iL^'be'p^cfeTrld. reason  already  given,  that  in  business  transactions,  we  do 
not  write  beyond  dollars  and  cents. 


AMERICAN    MONEY.  7o 

Example  1,  Divide  $9.67  by  5. 

OPERATION. 

5)9.67 

1.93  and  2  over,  which  fractionally  written  is  f, 
making  the  answer  $l.93|^.  l>ut  if  performed  so  as'  to 
have  millH  in  the  result,  it  would  be  d(inc  thus:  5)9.67,0 


1.93,4 
which  is  to  be  read  §1.93  cents,  4  mills. 

Note. — This  subject  will  be  treated  of  in  decimal 
fractions. 

2.  Divide  $535  by  17. 


OPERATION. 

17)535.00(31.47  iS 
51 


EXPLANATION. 

It  will  be  noticed  in  this  example, 

that  as  no  cents  w-crc  given  in  the 

sum,  two  ciphers  have  been  put  in 

25  1   the  cents  place;  while  in  the  (juo- 

17  tient,  two  figures  at  the  right  hand 

—  have  boon  marked  off,  showing  the 

80  .   answer  to  be  $31.47  and  the  frac- 

68  tional  expression,    fS^.     Had   three 

—  I   ciphers  been  annexed  to  the  divi- 

120  dend  instead  of.  two,  the  figure  on 

119  the  extreme  right  would  have  been 

mills. 

1  I 

3.  Divide  $25.44  by  16. 

4.  Divide  ^536  by  145. 

5.  Divide  $1000  into  250  equal  parts. 

6.  Divide  $6532.50  into  105  equal  parts. 

7.  If  1575  be  divided  equally  among  8  persons,  what 
will  be  the  share  of  each? 

8.  Bought  56yds.  of  straw  matting,  for  $20  :  what  was 
that  per  yard,  't 

9.  Hired  a  carpenter  for  a  month  of  26  working  days, 
at  $22  :  what  was  the  expense  of  his  services  a  tlay  ? 

10.  Sold  20  bags  of  Sea  Island  cotton  for  $1975  :  what 
was  the  worth  of  a  single-  bag  ? 

11.  At  $6  the  barrel,  how  much  flour  can  be  had  for 
$2.58  ? 

12.  At  75  cents   per   pound,  how  much   tea   can   be 
bought  for  $9  ? 

13.  How  many  barrels  of  apples    can  be   bought   for 
$45.50,  at  $3,-50  per  barrel? 


AMERICAN    MONEY. 

14.  Hew  long,  witli  the  wages  of  $1.12  J  a  day,  •■will  it 
take  a  laborer  to  earn  $40.50  ? 

A'ote. — In  this  exr.niplc  (Art.  107,  Note  Kx.  1),  beeaus*' 
C'f  tlie  fractional  >},  double  botli  divisor  and  dividend,  and 
theu  proceed. 

15.  For  C47.50,  bow  many  yards  of  broud  cloth  can  be 
had  iit  $2.37^- per  prd?  . 

16.  At  S7.''^0  a  ton.  how  many  tons  of  coal  can  be 
bought  for  >5255  ?     * 

17.  If  a  bag  of  coffee,  containing  IGUlbs.,  cost  $24 :  wiiat 
will  be  the  price  of  a  j,on,iil  ? 


MISCELLANEOUS  F.XAMPT.ER. 

Ici«  i^x.vr.iPLE  1. — Which  o...;..  ..,-.  j,;.,.^,,  _..j  Lusboli 
of  oiats  at  75  ceutte  per  bushel,  or  124  yards  of  calico,  at  7 
cents  per  yard  ?     What  is  the  difference  ? 

2.  What  is  the  difference  in  the  cost  between  5  toils,  of 
coal  at  ?;i7.60  per  ton,  and  12yds.  of  cloth,  at  $3.50  per 
yard. 

3.  How  many  acres  of  larid  at  o3  each,  may  be  bought 
with  the  value  of  46yds.  of  cassimdre  at  $3  per  yard  ?  ' 

4.  How  many  acres  of  land  ;Tt  $3.50  per  acre,  may  be 
purchased  with  the  value'  of  120hhds.  of  molasses  at  30 
cents  per  gallon. 

5.  Wlnit  is  the  cost  of  21b.  Ooz.  5pwt.  of  silver  at'  27 
cents  per  pwt.  ? 

JVot/i. — Ixeduce  to  pennyweights  and  then  multiply  by 
27.     This  gives  the  answer- in  cents. 

6.  What  is  the  value  of  8  tons,  9cwt  2qr.  ISlbs.  of 
sugar,  at  13  cents  per  lb.  ? 

7.  What  is  the  price  of  2  bushels  and  3  pecks  of  rice  at 
25  cents  a  peck  ? 

8.  What  will  51b.  7oz.  salts  come  to  at  9  cents  per 
lb? 


AMERICAN    MONEY. 

9.  A  mercliant  sold  a  remannt  of  ciotli  for  $25.50 ; 
thora  W'^re  Gyd-^. :  what  was  that  a  \ip.i\ '.' 

J,\//, , —  iU'iiuct'  Gyds   to   nails,   ace  _..".' 

cents. 

10.  VVliat  will  150  acres,  3rd.  18  perches  amount  to,  at 
;1.40  per  perch  ? 

11.  Bought  3  tous,  9cwt.  of  irou  for  $450 :  what  was 
ihat  per  CAvt.  ?  •  , 

12.  Tf,l  ton  of  hay  cost  $12^,  what  -will  5  cost? 

13.  If  1  lamb  cost  §2.25,  what,  at  the  .same  rate,  will 
15  cost? 

1-4.  If  85  bushels  of  corn  cost  §63.75,  what  is  that  per 
bushel? 

15.  If  a  butcher  purchase  17  beeves  for  §18Q,  25  sheep 
U)V  §(37.50  :  what  is  the  value  of  all '!  V'h.:\'i  of  the  beeves 
;  nd  sheep  each  per  head? 

10.  A  farmer  buys  3(3  sheep,  and  p:iys  for  them  with 
•'>  cows,  valued  at  §15  each,  ami  a  wagon  worth  §79  :  what 
lio  the  sheep  cost  each  ?  » 

17.  An  estate  valued  at  §21,000,  is  to  be  divided  amonu 
1  children,  when  the  widow  has  received  liet  portion  ul 
.>ue-third.     What  are  the  shares  ? 

18.  Bought  a  plantation  for  §4500 ;  and  paid  for  it 
with  10  shares  in  the  South  Carolina  Railroad,  §100   a 

liare ;  20  in  the  Savannah  Railroad,  at  §125  a  share: 
what  amount  will  be  called  fur  to  meet  the  '■•  '■■'••■  '^ 


76  COMPOUND    ADDITION, 


HIGHER     ARITHMETIC. 


PART   SECOND. 

COMPOUND  NUMBEES. 


A  fompound        132»  A  compound  number  consists  of  two  or  more 
number.  denominations  of  like  character,  as  expressed  in  the 

tables  of  currency,  weights  and  measures. 
What  are  com-     133«  Pounds,  shillings  and  pence  are  of  this  class, 
tcTi^.'^  "*"'"     ^^  before  stated ;  also,  dollars,  cents,  dimes,  bushels, 
quarts,  pecks,  etc. ;  but  these  or  any  other  denomina- 
tional values  cannot  be  united  to  form  a  common 
wnatarenot    ^'"^  •  thus,   it   is    impossible    to    say,  £5   and  $5 — , 
although    both   money  representatives — are    £10   or 
§10,  and  so  of  other  like  numerical  classes. 

134.  Ill  adding  comijound  numbers,  place  always 
addition  is  per- the  ditterent  values  or  measures  in  columns  oi  a 
formed.  Himilar  class,  and  in  the  order  of  the  tables,  com- 

mencing at  the  right  hand  with  the  lowest  value 
named.  When  set  in  order,  as  in  English  money, 
pounds  under  pounds,  shillings  under  shillings,  pence 
under  pence,  farthings  under  farthings,  add  the  right 
hand  column,  and  divide  it  by  the  number  of  this 
denomination,  to  make  one  of  the  next  higher :  the 
quotient  is  to  be  carried  to  the  next  column,  but  the 
remainder  placed  beneath  the  added  column. 


COMPOUND  ADDITION.       . 

135.  The  proof  is  the  same  as  in  simple  addition 
(Art.  39). 


COMPOUND     ADDITION. 


ENGLISH  MONF.Y. 


OPERATION. 

£ 

s. 

d. 

qr. 

20 

12 

4 

Example  1. — 15 

10 

6 

3 

14 

15 

10 

2 

10 

5 

9 

3 

25 

19 

8 

3 

85 

4 

7 

3 

Ans.     101 

IC, 

7 

2 

2 

3 

3 

EXPLANATION.  Explanation  of 

The  amount  of  the  sum. 
first  column  is  14qr. 
=3d.  2qr.  The  2qr. 
are  placed  in  the 
column  of  farthings, 
and  the  3d.  (the 
small  figure  below) 
carried  to  the  ponce 
column,  malcing 
43d.=3s.   7d.      The 


7d.  is  now  placed  in  the  pence  column,  and  the  3s. 
added  to  the  shillings  column,  making  56s.=£2  IGs. 
The  IGs.  is  placed  in  the  shillings  column,  and  the 
c£2  added  to  the  £,  making  the  answer  as  above. 

JSfote. — It  is  convenient  to  put  the  division  numbers 
above  the  columns,  and  the  carrying  ones  below  in 
small  figures,  as  in  above  example. 


£ 

5. 

d. 

qr. 

20 

12 

4 

30 

15 

9 

2 

16 

14 

8 

3 

17 

12 

6 

1 

19 

5 

4 

3 

3 

3 

10 

2 

Ans.    £87     12s.    3d.  3qr, 
2       3     2 


£ 

s. 

d. 

qr. 

4.  250 

17 

9 

2 

500 

16 

8 

3 

1250 

9 

0 

3 

25 

7 

5 

2 

35 

6 

4 

1 

£ 

s. 

d. 

qr. 

6.   55 

17 

10 

2 

16 

15 

9 

1 

14 

13 

1^ 

2 

5 

12 

5 

1 

29 

16 

3 

2 

d.  qr. 


3. 


12 

5 

4 

2 

5 

19 

11 

3 

16 

15 

4 

*? 

w 

18 

6 

9 

3 

7 

12 

8 

1 

£ 

s. 

d. 

qr 

36 

9 

0 

3 

19 

5 

10 

2 

45 

15 

11 

3 

120 

18 

9 

3 

16 

12 

8 

1 

£ 

5» 

d. 

qr 

20 

19 

10 

1 

85 

17 

9 

18 

12 

8 

3 

9 

14 

5 

o 

10 

15 

1 

3 

COMPOUND 

ADDITION. 

DRY 

MEASURE. 

hn. 

vh. 

qt. 

/>t 

bu. 

pk.  qt.  pt 

0 

O 

I 

1 

9. 

<iO 

3     6     1 

G 

2 

f; 

1 

.25 

2     7     0 

15 

t 

1 

12 

'  ."3     6     1 

25 

2 

5 

0 

5 

2     4     1 

LIQUID  MEASURE. 


tun.  hhd.  qal.  qt.pt. 
150     1-  ."45    2    i 

25    .8     87     ''     " 

55     1     20 

t.;;     2     15     -     r 
75     1     12     1     1 


11. 


tun.  hhd.  gal.  qt.  pt. 


5 

2 

25 

.:> 

1 

6 

15 

2 

1 

15 

.) 

^55 

o 

0 

9 

1 

bl) 

tr;* 

1 

7 

0 

25 

3 

1 

AVOIRDUPOIS  WEIGHT. 


tons.ctct.  qr.  lb.  oz.  dr. } 

12.  14  14  2  15  18  5  i 

15  3  24  14  3  1 

12  2  22  12  15  j 

9  2  15  10  0 

10  3  14  9  8 


to9is.ciDt.qr.ru.  oz.  dr. 

13.  22  17  2  20  15  14 

'   15  15  3  0  10  13 

5  12  2  8  9  12 

4  9  1  15  7  8 

16  10  2  22  5  7 


ArOTHECAEIES    WEIGHT. 


14. 


lb.  5. 

24  7 

17  11 

3G  6 

15  9 

9.  3 


5-  B-  gr. 
2  1  16 
7  2  19 
7 
13 
1   9 


5  0 
7  1 


15. 


lb. 
25 
15 
14 


11 

4 

2 

10 


y.  (/r, 


16 
15 
19 
11 

16 


TEOY  WEIGHT. 


.56. 


lb. 

45 

9 

11 

8 

10 


0  17   0 

3  14  ,  0 

0  3 

9  15 

0  10 


8 


0 
20 
17 
3  15 

0  16 


17 


lb. 
35 
40 
12 
6 
2 

5 
4 


oz.  pjivt.  qr. 

10  12  14 

9  15  13 

3  14  20 

7  8  10 

3  14  15 

9  17  IS 

10  18  20 


llhd. 

IS 
!:) 

14 

JS 
i9 


i'O.Mroij.Mj    ADDITION.   * 


ALE  AND  BEER  MEASURE. 


qal.  qt.  pt. 
"31     8     1 
1 


50 
47 
15 
12 
13 
3 


19. 


hhd. 


hd.  gal.  at.  p1. 
75    25    3     '^ 
3     15     1 


40 

50 


9 

1A     1 


0 

0 

2     0 


i'J.Fl.  or.  JUL 

]2-^     '■     :■ 


'&.E,  qr.  »< 


■111 


!.S)    :; 
4S    ;•] 


LONG  MEASURK. 


■  ;  • 

rur.po.  ft.  in 

210 

15 

5-15  10  2 

41. 

14 

3  .16  9  9 

9 

25 

3  20  14  2 

36 

12 

2  19  13  4 

16 

7 

1  7  12  2 

12 

5 

2  G  8  4 

.7  ^ 
'  'J- 

iili. 

I'"- 

i  1 . 

12 

9 

5 

9 

25 

31 

2 

5 

GO 

50 

3 

3 

19 

39 

2 

5 

14 

25 

3 

6 

G 

37 

'> 

8 

LAND  OR  SQUARE  MEASURE. 


sq.  yd.  sq.  ft.  sq.  in. 

97  .  4  104 

22    3  27 

105   • S  2 

37    7  IL'7 


sq.mi.  a.  r.  rd,  sq.iid. 

2fi.  2  60  3  37   2'5 

0  375  2  25   21 

7  450  0  31   20 

11  30  1  25   19 


CIRCULAR  AND  ASTRONOMICAL  MEASURE. 


")  17  36  29128. 

7  25  41  21 1 

8  15  10  09 


6  29  27  49! 29 
8  18  29  IG 

7  09  04  58 


5  25  25 
G  14  26 

7  08  0^ 


80 


•COMPOUND   SUBTRACTION. 


30. 


SURVEYORS    MEASURE. 


mi. 

fur. 

ch. 

rd 

•25 

8 

3 

10 

24 

5 

6 

09 

23 

4 

5 

8 

23 

3 

4 

7 

31. 


mi.  fur.  ch. 

35  '  5  2 

25      6  4 

23      7  5 

20      8  6 


rd. 
9 


SOLID    OR    CUBIC    MEASURE. 


cu.  yd.  cu.  ft.  cu.  in. 

32.   65    25  1129 

37    20  132 

50    1  1064 

22    10  17 


33. 


c.     eft. 
87   9 


26 
16 
19 


c. 
65 
35 
25 
15 


eft. 


TIME  MEASURE. 


34. 


mo. 

9 

<) 

o 

7 
6 
4 


3 
2 

1 

5 


,  2 
1 
2 
3 


yr.  da.  h.  "min.  sec. 

35.  89  59  20  13  12 

25  40  10  12  37 

5  90  19  19  25 

6  5  4  15  20 
5  6  14   5   6 


COMPOUND  SUBTEACTIOK. 

poirrid^Eubtr.ic-  ^S®'  '^^  ^^^  *^®  difference  between  compound  num- 
Don  is  to  be  bers,  jjlace  the  less  number  under  the  greater  of  sim- 
per o.me  .  1^^^  denominations,  and  beginning  at  the  right  hand, 
subtract  as  in  simple  numbers.  Should  tbe  figure  ia 
the  subtrahend,  or  lower  line,  exceed  the  one  above, 
add  to  the  number  in  the  minuend  as  many  as  it  takes 
of  that  denomination  to  make  one  of  the  next  higher, 
and  then  take  the  subtrahend  from  the  upper  figure 
or  minuend  so  increased.  Set  down  the  remainder, 
,!{\nd  carry  one  to  the  next  denominator  in  the  lower 
line,  and  bo  on. 


COMPOUND    SUBTRACTION. 


SI 


ENGLISH    MONET. 

Example  1. — From  45  pounds,  10  shillings,  6  pence, 
o  fai'things,  take  25  pounds,  9  shillings,  5  pence,  2 
farthings. 

£  5.         d.        qr. 

45        10        6  3 

.      25  9        5  2 


Ans.     £20 


Id.        Iqr. 


2.  Pi^om  35  pounds,  13  shillings,  7  pence,  2  farthings, 
take  25  pounds,  17  shillings,  10  pence,  3  farthings. 


OPBRATION. 

£       s.      d. 

qr. 

2(J         lli 

i 

35       13       7 

2 

29       17     10 

3 

EXPLANATION. 

In  this  example,  the 
upper 
less  tJian  the  lower,  p'" 
4  farthings=l   pen- 
ny, are  borrowed  and 


fi.  „   "  i^„- Explanation  o^ 
gure    being ^0*1^  in  «xi.m-, 


.471.'?.  £5  15s.  8d.  3qr 
added  to  the  2=6;  and  the  3  of  the  subtrahend  being 
subtracted,  the  remainder,  3^  is  set  down  in  the  far- 
things place,  and  Id. =4  farthings  that  was  borrowed, 
is  carried  to  the  next  figure,  104-1=11.  As  that  also 
('xceeds  the  number  of  the  minuend,  we  borrow  and 
add  12  pence  to  the  7=19,  and  say  11  from  19  is  8*; 
iind  setting  it  in  the  pence  place,  carry  the.  l8.=12 
pence  that  was  borroAved  to  the  next  figure,  17+1=18. 
As  tlvvt  exceeds  the  ujiper  figure,  we  borrow  and  add 
20  shillfngs  to  that,  and  say  18  from  33=15.  That 
being  set  in  the  shillings  place,  and  the  borrowed. 
1  =  2U  shillings  carried  to  the  pounds,  we  have,  after 
the  next  subtraction,  the  answer. 

In  a  similar  way  perform  all  examples,  taking  care 
to  operate  with  the  numbers  expressed  in  the  tables 
that  apply  to  the  proposed  sum. 


£ 

5. 

d 

qr. 

£ 

s.. 

d. 

qr 

From  50 

15 

4 

3 

4. 

From  37 

18 

9 

U 

Take  37 

14 

5 

2 

Take  18 

19 

5 

»; 

£ 

s. 

d. 

qr. 

£ 

■S. 

J. 

qr 

From  25 

19 

9 

2 

G. 

From  15 

6 

6 

•J 

Take     5 

16 

8 

3 

Take    14 

7 

6 

0 

82. 


COMPOUND    SUBTRACTION. 


DRY    MEASURE, 

hu. 
7.    From    38 

pk. 
6 

qt. 
3 

bu. 
8.  From   12 

f 

t 

Take    25 

4 

1 

Take      9 

7 

Mi' 

LIQUID    MKASCRK. 

hhd. 
9.    From   13 

'f 

qt.  pt. 
•0     0 

10.  From' 165 

gal 
15 

2     1 

Take     2 

39 

2     1 

Take     59 

36 

3     1 

AVOIRDUPOIS   WEIGHT. 


ion.  cwt.  qr.  lb.  oz.  dr 

11.  From  45  17  3  24  15  9 

Take     7  18  2  14  13  7 


ton.    cwt.  qr.  lb.    o»» 

12.  From  200     5     2  20  13 

Take     33  12     1  23  15 


APOTHECARIES     WEIGHT. 

lb.  o     3    B    gr,\                   lb-  3  o  9  gr. 

13.  From  3     15     0     14114.  From  4  0  4  0  0 

Take         5    2    7     171        Take  7  5  2  14 


TIME    MEASURE 
yr.  da.    h.  mm.  sec 

15.  From  95  89  16  15  15 
Take    75  57  23  30  17. 


yr.  da.    h.  min.  ace. 

16.  From  59     0    0    0    0 

Take    13     6  15.20  45 


Note. — 'In  subtracting  dates,  30  days  is  counted  a 
month. 


•Fo  &fld  the  dif-     17.  What  is  the  difference  in  time  betw'oen  March 
Ja^  "*       15^  1863,  and  January  13,  1869  ? 

yr.    mo   d. 

From    1869     1     13 

Take     1863    3     15 


Rem. 


9     2^ 


Note. — In  this  sum,  Januarj'  is  called  the  let  month, 
and  put  down  1 ;  March  the  3d,  and  put  down  3. 

18.  Calculate  the  time  from  July  6,  1857,  to  Decem- 
ber 9,  1860.     . 


COMPOUND    MULTIPLICATION. 


*3 


19.  Calculate  the  time  from  May  5,  1859,  to  August 
4,  1869. 

20.  Calculate  the  time  from  March  15,  1862,  to 
May  19,  1865. 

21.  IIow  long  is  it  from  September  9,   1861,  to 
November  4,  1870  ? 


COMPOUND  MULTIPLICATION. 

137.  In  Compound  Multiplication,  the  multiplier  iS)  The  moH.piie? 
without   exception,  an  abstract  number,  that  is,  a^mbcJ"'* 
number  separate  from  any  particular  object;  as  4,  9, 

15,  25,  etc.;  hence  the  product  is  simply  the  multipli- 
cand repeated  as  many  times  as  the  multiplier  ex- 
presses. 

138.  To  perform  compound  multiplication  :  having  to  pcrfonn 
placed  the  multiplier  under  the  lowest  denomination  oompoundmnt- 
of  the  sum,  multiply  this  and  divide  the  product  by 

the  number  it  takes  of  that  denomination  to  make 
one  of  the  higher;  having  set  down  the  remaindei', 
carry  the  quotient  to  the  product  of  the  next  de- 
nomination, and  so  Ofi. 


ENGLISH    MONKY. 


Example  1 


OPEKATIOHr. 

£     S. 
20 
—Multiply  5 .   7 
By 


explanation. 
d.  qr.  Here    we  E»pia»fciion  ot 

12     4       say,   first,    2*""*' 
9     2      |x6=12=3d. 
^      jAs    there    is 

: —       no  remainder 

Product,  £32    68  9d.  Oqr.jwe  write  0  in 
-     4     8       'the  farthings 


place  and  carry  the  ']d.  to  the  next  number,  after  its 
multiplication;  thus  9x0=54  and4-3=57;  this  divided 
byl2  gives  4H.and  9d.  over.  We  then  put  9d.  in  the 
pence  place,  and  carry  the  4s.  to  the  shillings  after  its 
multiplication ;  thus,  7  X  6=  VI  and+4=  46 ;  this  divided 
by  the  20  gives  £2  Os  over.  The  (Js  then  is  put  in 
the  shillings  place,  and  after  the  £5  has  been  multi- 
plied by  G=«30  the  2  is  added,  which  finishes  the  sum. 


84  COMPOUND    MULTIPLICATION. 

The  proof.  Multiplication    is  proved   by   division:    thus. 

6)£32  68.  9d.  Oqr.,  i# 

£b  Ts.  9d.  2qr  ,  the  operation  will  be  seen  in  Gom- 
pound  division  (Art.  140). 

LONG    MEASURE. 

rd.  yd.  ft.  in.  b.c. 

40  5J     3     12     3 

2.  Multiplicand,  15  4    2    6     1 
Multiplier,  4 

Product,  1  fur.  23     2     1     1     1 
16    0 


1  23  2  2  7  1 
In  this  example,  when  we  reach  the  denomination 
of  yards,  we  say  4x4=16  and  3  added  19;  by  the 
multiplication  of  this  by  2=38,  and  by  the  multipli- 
cation of  the  numbers  to  divide  it,  5 J  by  2=11,  11 
into  38=3  and  5  over;  5-^2,  to  bring  it  back  to  the 
right  denomination,  gives  2iyd. :  one-half  yard=lft. 
6in.  must  be  added  to  the  feet  and  inches. 


LIQUID 

gal.  qt.  pt.  gl 
3.  Multiply    5    3     12 
By                               6 

MEASURE. 

gal.  qt 
4.  Multiply    8     3 
By 

pt.  gi. 
1     3 

7 

1 

AVOIRDUPOIS    WEIGHT 

tons.     cwt.    qr.     lbs.      oz. 
5.  Multiply        14       17      3      20      10 
By 

dr. 

15 

4 

TROY    V 

lbs.   OZ.  pwt.  gr. 

6.  Multiply  50    7     14     19 

By                                  9 

HEIGHT. 

lbs.  OZ. 
7.  Multiply  25    7 
By 

pwt.  gr 

0    11 

8 

DRY    ME 

bu.  pk.  qt.  pt. 
8.  Multiply    30    3     6     1 
By                              10 

ASURS. 

bu,   pk.  qt. 
9.  Multiply      25     2    5 
By                             3 

COMPOUND  DIVISION. 


85 


ALR    AND    BEER   MEASURE. 


hhd. 

gal.  qt.pt.\ 

.  Mu 

Itiply  75 

37 

8     1 

By 

25 

Tihd.  gal.  qt.  pt. 

11.  Multiply  37     25     2     1 

By  15 


CIRCULAR  AND  ASTRONOMICAL  MEASURE. 


12.  Multiply 

By 


7  28  26  48 
20 


13.  Multiply 
By 


6  17  25  14 

6 


TIME    MEASURE. 

yr.      da.        h.  min.     sec. 

14.   Multiply      20       65       20  50       30 

By  12 


COMPOUND  DIVISION. 

139.  Compound  Division  is  the  process  by  'whicli  weco,„po„m)  ^jj. 
find  how  often  a  given  number  is  contained  in  a  dividend  vfeion  defined, 
of  compound  values  of  similar  nature. 

140.  To  perform  the  work,  divide  the  highest  denomi- 
nation by  the  given  number  and  set  down  the  quotient;  if  How  it  Ik  per- 
there  is  a  remainder,  reduce  it  to  the  next  lower  denomi-  *"^™*  ' 
nation ;  to  the  result  add  the  given  number  of  that  de- 
nomination and  divide  as  before;    and   so   on  until  the 

whole,  has  been  divided. 

Example  1.— Divide  £32  68.  9d.  Oqr.,  by  6  (Art.  138). 

EXPLANATION. 

•  £32-^6   gives   a  quotient   of  ^    ,     , 
£o,    and    a    remainder   ot    JtJ.  sum  iu  com- 
These   £2  reduced  to  shillings  p«"»'»  <ti'isio»- 
and  added  to  6s.  make  46s.,  which 
divided  by  6,  gives  a  quotient 
of  78.,  and  a  remainder  of  48. 
These  4s.  reduced  to  pence  and 
added  to  9d.,  make  57d.;  which . 


OPERATION. 

£ 

$. 

d. 

qr. 

20 

VJ. 

4 

6)32 

6. 

9 

0 

s.  £5 

78. 

9d 

2q 
6 

Proof,  £32    6s.  9d.  Oqr. 


m 


COMPOUND   DIVISION. 


divided  by  6  give  9d.  and  a  remainder  of  3d.  These  3d. 
reduced  to  farthings,  and,  as  there  are  no  farthings  to  be 
added,  divided  by  6,  give  2qr.,  which  finishes  the  work. 

ENGLISH    MONEY. 

2.  Divide  £474  17s.  9d.  2qr.,  by  4. 

3.  Divide  £90  15s.  7d.  3qr.,  by  9. 

AVOIRDUPOIS    WEIGHT. 

4.  Divide  46t.  17cwt.  8qr.  151b.  9oz.  7dr.,  by  10. 

5.  Divide  60t.  IScwt.  2qr.  201b.  6oz.  5dr.,  by  12. 

Note.- — When  the  divisor  exceeds  12,  it  is  necessary  t» 
put  down  the  work,  as  in  the  following  example  : 


DRV    MEASURE. 

6.  Divide  216bush.  3pk.  5qt.  Ipt.,  by  26. 

OPERATION. 

imh.  pk.  qt.  pt. 
4     8     2 
25)216    3     6     l(8bush.  2pk.  5qt.lApt- 
200 


16  bush. 
4 

67  pk. 
50 

17  pk. 


141  qt. 
125 

16  qt. 

9 


33  pt. 
25 


EXPLANATION. 

Finding  that  25  is  contained  8 
times  in  216,  we  put  it  in  the. 
q-do*ient  and  multiply  25  by  8== 
200  :  this  we  subtract  from  216 
and  have  a  remainder  of  16  bush- 
els. These  reduced  to  pecks  make,, 
with  the  three  in  the  sum,  67 
pecks.  Dividing  by  25,  aad  put- 
ting the  2,  which  are  the  peeks^ 
in  the  quotient,  we  multiply  the- 
divisor  by  the  2  and  subtract.  The 
17  that  remain  are  pecks,  and 
being  reduced  to  quarts,  we  divide 
as  before,  and  so  on. 


APOTHECARIES    WEIGHT. 


7.  Divide  4561b.  6J  23  19  18gr.,  by  49. 

8.  Divide  1251b.  bl  \Z  29  15gr.,  by  36. 


COMPOUND   DIVISION.  ST 

CLOTH  MEASURE. 

9.  Divide  45yd.  3qr.  Sua.,  by  15.  , 

10.  Divide  85yd.  2qr.  2na.,  by  9. 

LIQUID  MEASURE. 

11.  Divide  149hhd.  45gal.  3qt..2pt.,  by  50. 

12.  Divide  TOlihd  25gal.  2qt.  2pt.,  by  8. 

LONG  MEASURE. 

13.  Divide  34deg.  24m.  7fur.  20po.  lift.  9ih.,  by  30. 

14.  Divide  420deg.  37m.  4fur.  30po.  5ft.  3in.,  by  65. 

LAND  OR  SQUARE  MEASURE. 

1-5.  Divide  978q.  yd.  4sq.  ft.  l04sq.  in.,  by  33. 

16.  Divide  80  sq.  yd.  3sq.  ft.  97sq.  in.,  by  26. 

TROY  WEIGHT. 

17.  Divide  61b.  15oz.  20pwt.,  by  9. 

18.  Divide  a21b.  ISoz.  lopwt.,  by  35. 

ALE  AND  BEER  MEASURE. 

19.  Divide  12hhd.  50gal.  3qt.,  by  6. 

20.  Divide  3hhd.  47gal.  2qt.,  by  22. 

DRY  MEASURE. 

21.  Divide  6bush.  3pk.  7qt.  Ipt.,  by  7. 
23.  Divide  12bush.  3pk.  Gqt.  Ipt.,  by  18. 

CIRCULAR  AND  ASTRONO.\fICAL  MEASURE. 

23.  Divide  lOcir.  25s.  30'  40",  by  9. 

24.  Divide  9cir.  20s.  18'  30",  by  33. 

surveyor's  MEASURE. 

25.  Divide  12m.  7fur.  9cb.  3rd.  201i.,  by  6. 

26.  Divide  25ra.  6fur.  8ch.  2rd.  151i.,  by  25. 

SOLID  OR  CLDIC   MEASURE. 

27.  Divide  4c.  14e.  ft.  25t.  24c.  yd.  UOOcu.  ft.,  by  9. 

28.  Divide  5c.  15c.  ft.  39t.  25c.  yd.  1650cu.  ft.,  by  28. 

TIME  MEASURE. 

29.  Divide  25yr.  335da.  21h.  45min.  55scc.,  by  6. 

30.  Divide  50yr.  260da.  19h.  35min.  403ec.,  by  36. 


88  PROPERTIES    OF   NUMBERS. 


PROPERTIES  OF  NUMBERS. 

Prime  and  com-      141.  Numbers  are  of  two  kinds,  prime  and  composite. 
C  «um-         JI2,  Ajyrime  is  always  divisible  by  itself  and  1,  but  can 
A  prime  num-  j^e  divided  by  no  other  whole  number  :  ],  2, 3,  5, 9,  13,  etc., 

are  of  this  order. 
A  composite         143.  A  comjtosite  number  is  one  fhat,  while  divisible  by 
xumber.  itself,  can  be  divided  by  other  numbers :  such  are  4,  6, 

8,  15,  etc. 
Numbers  con-       144.  Number s,  whether  prime  or  composite,  are  concr«?/e 
sTraV*^*'"    ^rid.  abstract. 

\.  concrete  145.  A  concrete  number  is  that  which  is  applied  to  a 

mtmber.  particular  object;  as  when  is  said  5  bushels,  12  books,  20 

years. 
An  abstract  146.  An  abstract  number  stands  disconnected  with  any 

number.  ^^j^^^^  ^^^^  j^  ^j^^pj^.  3^  4^  5^  ^0,  etc. 

A  further  disr       j^^j^  Composite  numbers  are  ^e//ec<  or  t'mper/ec^' 

posite  nmn-  148«  A  perfect  number  (of  which  nine  only  are  known) 

"TpOTfectuum  is  equal  to  the  sum  of  all  its  integral  factors;  thus,  28  is 

her.  of  this  class,  for  the  sum  of  1+2+4+7+14=28. 

An  imperfect        149.  An   imperfect  number  is  not  equal  to  the  suiii  of . 

■umber.  \^  factors;  thus,  8  is  an  imperfect  number,  for  the  sum 

of  its  factors,  1+2+4,  is  not  equal  to  8. 
What  are  fac-      1^0.  The  factors  of  a  number  are  the  makers  or  produ- 
t«r3.  cers  of  it ;  thus  5  and  6  are  the  factors  of  30  ;  3  and  8,  or 

3  and  4  and  2  are  the  factors  of  24. 
Prime  factors.       151.  The  ptrime  factors  of  a  number  are  the  lowest  figures 

whose   continued   product  make  the   number ;    thus,  the 

prime  factors  of  24  are  2,  2,  2  and  3 ;  the  prime  factors 

of  36  are  2,  2,  3  and  3. 
wiiat  are  not       152.  Prime  numbers  being  divisible  by  themselves,  arc 
prime  factors,  ^qi  called  prime  i'actors. 

An  intc'or  de-      153.  An  integer  or  whole  number  is  a  unit,  or  a  collec- 
fincd.    °  tion  of  units. 

When  a  work  is      154.  When  both  thc  divisor  and  quotient  are  integers, 
called  exact,     ^j^g  ^qj-Jj-  jg  called  cxact. 
As  aiiqijot  part.      155.  A  number  contained  in  another  an  exact  number  of 

times  is   known  as   an  aliquot  part;   thus  10   cents,  12J 

cents,  33i  cents,  and  50  cents,  are  aliquot  parts  of  a  dollar. 
A  measure  of  156.  A  measure  of  any  quantity  is  contained  in  that 
any  quantity:    quantity  a  certain  number  of  times  without  remainder; 

thus,  3  is  a  measure  of  6,  and  8  of  24. 


PROPERTIES   OP   NUMBERS.  8!^ 

157.  The  common  measure  of  two  or  more  numbers  is  The  oommoK 
any  number  that  will  divide  each  of  them  without  remain-  "''^"''•"'"e. 
der  ;  thus,  4  is  a  common  measure  of  12,  IG,  24. 

158.  The  qrcatcst  common  meaxurc  of  two  or  more  num- Tho  Rreatest 
bers  IS  the  greatest  number  tlvat  wul  measure  each  or  them ;  uro. 
thus,  6  is  the  greatest  common  measure  of  12,  18  and  30. 

A  measure  is  sometimes  called  a  sub-multijilc.  a  snivniukipii?. 

159.  The  multiple  of  any  (juanUtij  contains  that  quantity  ^  ,„„itipieGf  a 
a  certain  number  of  times  without  remainder;  thus,  20  is  quanuty. 

a  multiple  of  5,  and  18  of  3. 

160.  A  common  multiple  of  two  or  more  numbers  is  any  j^^,,j,n,„^,m,n,;. 
Tiumber  that  may  be  divided  by  each  of  them  without '■'p'o- 
remainder;  thus,  48  is  a  common  multiple  of  4,  6  and  8. 

161.  The  least  common  multiple  of  two  or  more  numbers  xhc  u-Ai't  cow- 
is  the  least  number  that  is  exactly  divisible  by  each  of  the  '^"^"  miiitipic 
given  numbers ;_  thus,  24  is  the  least  common  multiple  of 

4,  6  and  8. 

162.  The  reciprocal  o£' a,  number  is  the  quotient  arising  The  rceiprooain 
from  the  division  of  a  unit  by  that  number;  thus,  the  °^  ** '"""^'^'■• 
reciprocals  of  4,  9  and  12,  are  \,  ^,  f^^. 

PRIME    FACTORS. 

163.  To  find  the  prime  factors  of  a  number,  divide  the  To  fmd  th" 

a  umber  by  the  least  prime  number  that  gives  an  exact  quo-  P''""'^  «*ctois. 
tient ;  then  divide  that  quotient  by  the  least  prime  number 
as  before,  and  so  on.     The  several  divisors  and  the  last 
quotient  will  be  the  prime  factors  of  the  given  number. 

Example  1. — What  are  the  prime  factors  of  42  ? 

OPERATION. 

2)42 
3)21 

7X3X2=42. 

2.  What  are  the  prime  factors  of  30  ?    Ans.  2,  3  and  5. 
8.  What  are  the  prime  factors  of  56  ? 

4.  What  are  the  prime  factors  of  63  ? 

5.  What  are  the  prime  factors  of  76  ? 

6.  What  arc  the  prime  factors  of  180  ?     Ans.  2,  2,  3,  3 
and  5. 

7.  What  are  the  j)rime  factors  of  7684  ? 


M^^  GREATEST   COMMON   DIVISOR. 


GREATEST  COMMON  DIVISOR. 

To  find  the  164.  To  find  the  greatest  common  divisor  or  measure  of 

mon^dwieor""  ^^0  or  more  numbers :  resolve  each  number  into  its  prim* 

factoi*s,  and  the   product  of  the  factors  common  to  each 

result  will  be  the  answer. 

Example  1. — What  is  the  greatest  common  divisor  of 

18,  30  and  48? 


OPERATION. 

18=2x3x3 
3U=2x3x5 
48=2X3X2X2X'2 


OPERATION. 

60=2X2X3X5 
72=2x2x2x3x3 
48=2X2x2x2x3 
84=2x2x3x7 


Here,  it  is  seen  that  2  and  3  ar« 
factors  common  to  all  the  numbers, 
and,   also  that   they  are   the   only 
common  factors;  hence,-  their  pro- 
duct, 2x3=6,  is  the  greatest  common  divisor. 

2.  What  is  the  greatest  common  divisor  of  60,  72,  48 
and  84  ? 

Here  2  is  a  factor  more  than 
twice  in  some  of  the  given  numbers, 
but,  as  it  is  a  factor  only  twice  in 
others,  we  can  only  take  it  twice  to 
find  the  result. 
3    What  is  the  greatest  common  divisor  of  24,  48  and  96? 

4.  W.hat  is  the  greatest  common  divisor  of  25,  75,  90, 
85,  too  and  125?     Ans.  5. 

5.  What  is  the  greatest  common  divisor  of  48,  72,  120, 
144  and  168  ? 

What  is  to  be     When  the  numbers  are  not  readily  the  subject  of  the f 
.jono  when  the  rule   given,  divide  the  greater  number  by  the  less,  and 
"uidc.*""*  *     the  first  divisor  by  the  remainder,  and  so  on,  until  there  io 
no  remainder;  the  last  divisor  will  be  the  answer.  . 

6.  W^hat  is  the  greatest  common  divisor  of  28  and  40  ? 
Ans.  4c. 

OPERATION. 

28)40(1 
28" 

12)28(2 
24 

4)12(3 

7.  What  is  the  greatest  common  divisor  of  16  and  28  ? 
Ans.  4. 

8.  What  is  the  greatest  common  divisor  of  8  and  16  ? 
Ans,  1, 


THE  LEAST   COMMON    MULTIPLE.  ^^ 

9.  What  is  the  greatest  common  divisor  of  21G  and 
408  ?     A*if!.  24. 

10.  What  is  the  greatest  common  divisor  of  315  and^ 
810? 


THE  LEAST  COMMON  MULTIPLE. 

165.   To  find  the  least  common  multiple  or  dividend  of  To  find  the 

1  .1  •  1       •         1   1  r    „   least  coiiiT«i>"i» 

two  or  more  numbers:  arrange  these  in  a  horizontal  line, ^jyitij^g, 
and  divide  by  anj  prime  number  that  will  go  into  two  or 
more  of  them  exactly,  and  place  in  a  lower  line  the  quo- 
tients and  undivided  numbers.  Proceed  in  the  same  man- 
ner until  there  is  no  prime  number  greater  than  1  that 
will  divide  without  remainder  any  two  of  the  numbers. 
The  figures  beneath  the  lower  line  and  the  divisors  multi- 
plied together  will  give  the  answer. 

Example  1. — Find  the  least  common  multiple  of  3,  4 
and  8.     Ans.  24. 

OPERATION. 

2)3 4 8 


2)3. 


3 1 2 

3x1x2x2x2=24. 

2.  What  is  the  least  common  multiple  of  6,  8,  12,  18 
and  24  ? 

3.  What  is  the  least  common  multiple  of  33,  44  and' 
55? 

4.  What  is  the  least  common  multiple  of  9,  11,  17,  19 
and  21  ?     A71S.  223839. 

5.  What  is  the  least  common  multiple  of  4,  14,  28  and 
98? 

6.  What  is  the  least  common  multiple  of  2,  7,  5,  G  and 
8? 

7.  What  is  the  least  common  multiple  of  4,  12,  20  and 
24? 

8.  What  is  the  least  common  multiple  of  2,  7,  14  and 
49? 


92  MISCELLANEOUS    EXAMPLES. 


MISCELLANEOUS  EXAMPLES.  » 

168.  Example  1.— From  £2  10s.  take  17s.  9d.  3qr.      • 

2.  From  an  ingot  of  silver,  weighing  51b.  3oz.  15dwt., 
six  silver  spoons,  each  in  weight,.  2oz.  lOdwt.  12gr.,  were 
made  :  what  remained  ? 

3.  From  some  merchandize,  weighing,  18t.  lOcwt.  3qr., 
were  sold,  15t.  r4cwt.  2qr.  141b. :  what  was  left? 

4.  A  merchant  had'  455t.  of  sugar,  but  sold  225t. 
16cwt.  3qr.  201b. :  what  was  over  ? 

5.  A  merchant  sold  from  a  piece  of  broad  cloth,  con- 
taining 30yd.  3qr.,  16yd.  2qr.  3na. :  whal  quantity  had  he 
left? 

6.  An  importer  of  wine  sold  from  an  invoice  of  38  tun, 
Ihhd.  ISgal.  2qt.  Ipt.,'l7  tun,  3hhd.  42gal. :  how  much 
had  he  left  ? 

7.  From  50hhd.  34gal.  2qt.  of  ale,  were  sold  29hhd. 
36"gal.  3qt. :  what  remained  ? 

8.  A  .planter  divided  a  plantation,  containing  869 
acres,  3rd.,  into  5  parts  :  what  was  in  each  ? 

9.  What  is  the  10th  part  of  lyr.  3mo.  2w? 

10.  What  is  the  5th  part  of  20h.  49m.  50sec? 

11.  If  a  bale  of  English  cloth,  containing  356yd.,  cost 
£3425  2s.  4d. :  what  is  the  pi-ice  per  yd.  ? 

12.  Bought  40  loads  of  wood,  each  measuring  Ic.  3c.  ft. 
7cu.  ft.,  at  $4.50  per  cord :  what  was  the  entire  quantity 
and  cost? 

13.  If  a  ship  sail  2°  15'  29"  a  day:  what  will  she  have 
sailed  in  25  days  ? 

14.  If  five  laborers  dig  a  ditch,  6rd.  5ft.  deep,  in  3 
days,  what  will  they  have  done  in  15  days  ? 

15.  What  will  61b.  75  53  29  lOgr.  come  to,  at  5  cents 
per  grain  ? 

16.  What  is  the  cost  of  2t.  12cwt.  3qr.  201b.  of  sugar, 
at  15  cents  per  lb.? 


VULGAR   FRACTIONS.  93 


FRACTIONAL  ARITHMETIC. 


PART    THIRD. 

VULGAR  FRACTIONS. 


A  fraction  dfl- 

167.   A  fraction  is  one  of  the  equal  parts  of  a  unit,  or 
a  collection  of  units.  w^y  Tulgar 

•ir»o     rni        c         •  i  ■        .  ,  .  tractions,  and 

loo.   ine  traction,  under  present  consideration,  called  how  expressed; 
vulgar,  from  being  in  most  common  use,  is  expressed  by  two 
numbers,  vertiqally  placed,  with  a  separating  line ;  thus, 
i-  one-half;    ii   two-thirds ;    |   three-fourths. 

169.  The  number  that  denotes  into  how  many  parts  the  ^r'^d^K"'"* 
unit  is  divided  is  called  the  denominator.     Its    place  is  ' 
always  below  the  line.     Thus,  in  the  fraction  i,  2  is  the 
denominator,  and  it  denotes  that  1  has  been  separated  into 

two  parts.*  • 

170.  The  figure  above  the  line,  as  it  names  or  numerates 

(that  is  numhers)  how  many  parts  of  a  Unit,  when  divided  Thn  numerator 
are  used,  is  called  the  numerator  •  thus,  in  the  expression  '^'^^^^'^' 
^,  one  part  of  the  two  parts  into  which  the  unit  was  sepa- 
rated is  denoted  to  be  used. 

17L   Taken  together,  the  numerator  and  denominator  are 
called  the  terms  of  the  frac'ion.  Terms  of  a  frao- 

Uon. 

Note. — For  convenience  of  expression,  we  always  say 
numerator  and  denominator. 

172.  A  fraction  is  simply  a  peculiar  form  of  writing  a  Fractions  aro 
divisor  and  dividend,  the  denominator  being  the  divisor,  Kk^™^ 
and  the  numerator  the  dividend ;  thus  -}  is  the  same  as 
0-^3=3  ;  or  J  is  the  same  as  1-4-2=  i.     Here  the  value  of 
the  fraction  is  seen  to  be  the  quotient  of  the  numerator 
divided  by  the  denominator. 


^4  VULGAR  TRACTIOKS. 

A   iro  er  fiae-     ^'^^'  Fractions  are  variously  known.     A  proper  fraction 
tion"  ^  has  a  less  number  for  its  •  numerator  than  for  its  denomi* 

nator,  as  -f ,  f.  ,         i.       • 

An  improper        174.  An  improper  fraction  has  either  equal  numbers  in 
fraction.  j|^g  terms,  thus,   f ;    or,   is   greater  in  its  denominator ; 

thus  -^  •^^' 
A  simple  frac-     Yib!' A  simpU  fraction  \^  that  which  has  whole  numbers 
tion.  in  }3oth  numerator  and  denominator  ;  thus,  i,  -f-. 

A  componnd        176.  A  compound  fraction  is  a  fraction  of  a  fraction,  or 
ira.;tion.  ^^^  fractions  connected  by  the  word  .of;  thus,  ^  of  -f. 

177    A  mixed  number  is  a  whole  number  and  a  fraction 
Admixed  num-^^.^^^  .^  ^^^  ^^^.^  .    ^^^^^^  5^^  jgj^ 

178.  A   Complex  fraction   is   one   whole  numerator  or 
fra^tiolu'^''       denominator,  or  both,  and  has  a  fraction  or- mixed  number  in 

51    ^  4    9t 
one  or  each  of  its  terras ;  thus,  -_  4.  — -  — 

6      0   oj  o4 

179.  Kemark. — As  the  addition  and  subtraction  of  frac- 
tions involves  some  preliminary  work,  we  commence  with 

.  examples  in  v  ultiplica/ion. 
■10  muKipiy  a      Example  1.— Multiply  f  by  3. 

traction  by  o  OPEUATION. 

^liole  number.  f  X3=-f-  Ajl^. 

It  will  be  seen  that  the  work  is  done  by  the  multipliea- 
•  tion  of  the  numerator  by  the  whole  number. 

2.  Multiply  i  by  3. 

3.  Multiply  i  by  2. 
I                            4.  Multiply  i^s  by  4- 

5.  Multiply  h  by  *.  Ans.  ■fj=6. 

6.  Multiply  iS  by  8. 

7.  Multiply  2^5  by  7. 

8.  Multiply  i-o  by  6. 

9.  Multiply  s  by  5. 

10.  Multiply  f  by  7.  t  1    • 

Sometimes,  when  the  numbers  are  larger,  the  work  ib 

Biraplified  by  dividing  the  multiplier  by  the  denominator, 

and  multiplying  by  the  numerator;   but,  on  the  whole, 

the  former  method  is  preferable. 
.11.  Multiply  i  by  25.     Ans.  \\ 
By  the  division  of  the  denominator -J- of  2o=^^y.?'=^ 

=8i. 

12.  Multiply  t  by  Sn. 

13.  Multiply  ^  by  53. 

14.  Multiply  ft  by  (35. 

^Vo<e. — When  the  numerator  exceeds  the  denominator, 
divide  by  the  denominator. 


VULGAR   FRACTIONS.  U5 

180.  Example  1. — Multiply  3  by  i.  Tomnuipiyor.e 

OPERATION.  anolhe"?-  ^^ 

5X1=1^  Ans. 
This  is  (lone  by  multiplying  the  numerators  together  for  a 
new  numerator,  and  the  deuonunators  together  for  a  new 
denominator. 

2.  Multiply  f  by -f^j. 

3.  Multiply  if  by  if. 

4.  Multiply  -?i  by  ^. 

5.  Multiply  //s-  by  if.         ' 

6.  Multiply  .,W  by  fg. 

7.  Multiply  ^  by  f. 

8.  Multiply  n  by  A. 

9.  Multiply  if  by  -3^. 
10.  Multiply  ^X  by  n- 

EXAMPLES  IN  DIVISI6K. 

181.  Example  1.— Divide  fj  by  6.     , 

H^  OPEPvATION.     • 

Here  the  numerator  is  divided  by  the  whole  number,  and  tion  by iT ?.h^e 
the  same  denominator  is  placed  beneath.  number. 

2.  Divide  ff  by  6. 

3.  Divide  iJ  by  13. 

4.  Divide  /rV  by  9.' 

5.  Divide  -3^3^  by  6. 

6.  Divide  rW  by  9. 

7.  Divide  3^^  by  11. 

8.  Divide  i|f  by  12. 

9.  Divide  i'j^  by  15. 
10.  Divide  i?a  by  20. 

Remark. — In  most  instaneos,  the  result  required  is  Muiupiying,  jm 
more  easily  obtained  by  multiplying  the  denominator  by  somiMusunce*, 
the  whole  number;  but,  usually,  the  division  of  the  tof. '^"'°"'*""'" 
numerator  is  preferable. 

182.  ExAMPLK  1.— Divide  i  by  f . 

OPERATION'.  „,       ..   .   .  , 

tXt=^6='!J8-  Ans.  a  iVaction  hy  ;( 

In  this  case,  we  invert  the  divisor,  and  multiply  tho  ^'"'*^'''°"' 
Bumeraturs  and  <l('nominators  together. 

2.  Divide  -]%  by  \*^. 

3.  Divide  -f *-  by  if-. 

4.  Divide  ^|  by  ^. 

5.  Divide  |-|  by  -j4- 

6.  Divide  il  by  VV- 


96  VULGAB  PEACTI0N8. 

7.  Divide  if  by  A- 

8.  Divide  f^  by  i. 

9.  Divide  ff  by  f . 
10.  Divide  -H  by  -f . 

REDUCTION. 

183.  Example  1. — Reduce  -ff  to  its  lowest  terms. 


To  reduce   a 
fraction  to  its 


OPERATION. 


lowest  terms.  -J-2_^'3— =A.  j^ng. 

Here,,  the  terms  of  the  fraction  are  divided  by  their 
greatest  common  divisor,  3.  Sometimes  repeated  divisions 
are  employed  before  the  final  result  is  obtained. 

2.  Reduce  -fl  to  its  lowest  terms. 

3.  Reduce  -|f  to  its  lowest  terms. 

4.  Reduce  -^-  to  its  lowest  terms. 

5.  Reduce  -1^  to  its  lowest  terms. 

6.  Reduce  -f^  to  its  lowest  terms. 

7.  Reduce  -fg-  to  fts  lowest  terms. 

8.  Reduce  ff  to  its  lowest  terms. 

9.  Reduce  ^%  to  its  lowest  terms.  ^ 

In  this  example,  it  will  be  noticed  that  5  is  coilimon  to 
both  terms,  and  that  t^7'V-t-5=-3  5- J  it  will  be  also  noticed 
that  7  is  common  to  both  the  terms,  obtained  by  the 
division  of  5,  and  that  i-i-4-7=f ,  the  answer. 

10.  Reduce  -^s%\  to- its  lowest  terms. 

11.  Reduce  iVsV  to  its  lowest  terms. 

OPERATION. 

-,^,^dbl=i  Ans. 

12.  Reduce  t^Vs  to  its  lowest  terms. 

184.  Example  1. — Reduce  ■^-  and  V-  to  their  equiva- 
lent whole  or  mixed  number. 
To  reduee  an  OPERATION. 

improper  frno-  96-7-6=16  ;  and  54-4-8= 6f  Ans. 

tion  to  a  whole  ' 

or  mixed  Bum-jJere,  the  numerators,  as  being  in  excess  of  the  denomi- 
nators, are  divided  by  them  for  the  required  result. 

2.  Reduee  ^^  to  a  mixed  number. 

3.  Reduce  ^f-  to  its  whole  number. 

4.  Reduce  -^g*  to  a  mixed  number. 

5.  Reduce  -^f-P-  to  a  mixed  number. 

6.  Reduce  -^i^-^^  to  a  mixed  number. 

7.  Reduee  -V^  to  a  mixed  numlier. 

8.  Reduce  ^^  to  a  whole  number, 
185,  Example  1. — Reduce  4-?- to  4ths. 

OPERATION. 

4x4=164- 3=Y  ^ns. 


VULGAR    PUACTIONS.  97 

Here,  the  whole  number  is  multiplied  by  the  denominator  To  reduce  Va* 
of  the  fraction,  the  numerator  added  to  the  result,  and  a  y"'^®**  number 

o       .•  1        ,  ,      .  ,         ,  ,  '  to  an  imprbper 

new   traction    made,   by  placing  the  denonnnator  given,  fraction.       _i,i 
under  the  sum  obtained  as  above. 

2.  Reduce  65  J  to  8th,s. 

3.  Reduce  35-6-  to  9ths. 

4.  Reduce  40t^o-  to  lOths. 

5.  Reduce  bZ^\  to  12ths. 

6.  Reduce  875f  to  au  improper  fraction. 

7.  Reduce  7548^-^  to  an  improper  fraction. 

8.  Reduce  5690|-  to  an  improper  fraction. 

9.  Reduce  3501^^  to  an  improper  fraction. 
10.  Reduce  675^  to  an  improper  fraction. 

186.  Example  1. — Reduce  5f 

6f  to  a  simple  fraction. 

OPERATION. 
•^3 —  5  To  roducc'.'a. 

6i=Y=V^-^J   -»fXTiV=T¥»  ^»i«-  complex  frac- 

^_  1  «7j^7i3»  tion  to  a.  .Simple 

Wc  here  reduce  the  numerator  and  denominator  of  the  '^"<'- 
complex  fraction,  each  to  a  simple  one ;  and,  inverting  the 
terms  of  one  of  the  fractions,  multiply  numerators  and 
denominators. 

2.  Reduce  8-|- 

15f  to  a  simple  I'ractiou. 

3.  Reduce  3-j^ 

5-f-  to  a  simple  fraction. 

4.  Reduce  6 J 

81  to  a  simple  fraction. 

5.  Reduce  -^ 

9|-  to  a  simple  fr.iction. 

6.  Reduce   5-| 

\  to  a  simple  fraction. 

7.  Reduce  45 

-?  to  a  simple  fraction. 

8.  Reduce     f  to  a  simple  fraction. 

OPERATION, 

Note.— A  whole  numb.M-  is  divided  by  a  fraction,  by 
multiplying  the  Avholo  nuinbcr  by  the  denominator, 
and  then  dividiiiir  tbe  product  by  the  numerator. 


98  VULGAR   FRACTIONS. 

9.  Reduce  -f-  to  a  simple  fraction. 

10.  Reduce  f  to  a  simple  fraction. 
Toireduce  frae-     187.  EXAMPLE  1. — Reduce  f,  |  :iud  +  to  a  common 
♦ioBB  to  a  com-  denominator. 

mon  denomina- 

rwr.  OPERATION. 

8x8x5=120,  1st  numerator, 
5x5x5=125,  2d  numerator, 
4x5x8=160,  3d  numerator, 
5x8x5=200,  common  denominator; 

X2.A     12  6.    Xfi-Q.       J„<t 
2  0  0)    2  0  U>    2  0  (» •     -T-«*. 

Here,  we  multiply  the  numerator  of  each  fraction  by 
all  the  denominators,  except  its  own,  for  the  new 
numerators ;  and  all  the  denominators  together,  for 
a  common  denominator. 

2.  Reduce  |,  f-  and  -I-  to  a  common  denominator. 

o.  Reduce  'i,  f  and  f  to  a  common  denominator. 

4.  Reduce  f ,  ^}  and  4  to  a  common  denominator. 

5.  Reduce  -I,  -^,  -^f  and  f  to  a   common  denomi- 
nator. 

6.  Reduce  j-,  yV,  -f  and   if  to  a  common   denomi- 
nator. 

7.  Reduce  J,  f ,  -f  and  ^  to  a  common  denominator. 

8.  Reduce  -fV,  f,  f  and  f  to  a  common  denomiaator. 

9.  Reduce  7^,  |  and  5  to  a  common  denominator. 

JVote. — When  there  is  necessity,  as  in  this  example, 
reduce  to  simple  fractions. 

10.  Reduce  42, 5-f  and  1 J  to  a  common  denominator. 
To  reduce  frac-     188.  EXAMPLE  1. — Reduce  i,  t^,  -j^  and  i-}-  to  their 

ttons  to  the     least  oommon  denominator. 

least  common 

»iu!tiple.  OPER.vnOX. 

iij-f     -N    -f-i    ii  I  ■^fi^x3=27,  1st  numerator. 

■ffx5=30,  2d  numerator. 

x|x7=28,  3d  numerator. 

4fX  11=33.  4th  numerator. 

2x2x2x3x  1x1=72,  least cora- 
1  ^5  1  uioii  multiple  of  denominators. 
lIen<-^>,  f,  i\, -?aMd^i=|-z-,f8-,  4A;uidf|.  Ans.  Here, 
as  the  i'ractio.is  are  in  the  lowest  terms,  we  find  the 
least  common  multiple  of  the  denominators  for  a 
<-ommon  denominator.  Then.  dividini<;  this  by  each 
denominutoj",  we  'niidtiply  each  (luoiient  by  its  owtt 
numerator. 


2)4 

6 

9     12 

2)2 

3 

9       6 

3)1 

't 

!»       3 

\  ll(;ak  tbactioxs. 


90 


1'.    Reduco  f,  -i\,  f  and   \   to  n  coimnon  deuomi'i:)- 


lor 


-fr-  of  a  >hi!HiiM-  to  il\ 


ToZrcditi'o  ;i 
fraction    of  » 
hichor  to  on*  of 
a  terror  denom- 
ination. 


:{.  Reduce  -i^j-,  tV,  -/g-  and  VV- 

4.  Reduce  i,  -,V,  -A-  and  -/--. 

5.  Reduce  f,  -gV.  H  and  -iV- 
0.   Reduce  -,V,  ii,  -A-  and  -3^ 

y4«c-      .14  7  0     X?,2J>- 
-'*■''*•     3  15  0'    31  5  U ; 

7.  Eoduce  3,  4,  -f  nnd-iV- 
S.  Reduce  -,V>  i<  i  and  ;. 
0.  Reduce  ■?,  -iV-  -/o-  and  -jV- 
10.  Reduce  MJ,  i  and  -§-. 

189.  ExAMPLK  1.— Re.iuci 
fraction  ol'a  i'ai'lliing. 

orEKATION. 

AYe  multiply  the  fraction  to  be  reduced  by  S'tch 
numbers  as  are  neceH><arv  to  the  result. 

2.  Reduce  -^-o  of  a  ]iouiKi  Troy,  to  tlie  fraction  of 
H  grain.  Ans.  ^-^.  t 

o.  Reduce  -g-J-o   of  a  pound  Apothecaries'  weighl, 
to  the  fraction  of  a  ixrain.  A7(S.-^-^. 

4.  Reduce  -^^^-5  of  a  day  to  the  fraction  of  a  second. 

5.  Reduce  f  ol'a  hJul.  to  the  fraction  of  a  quart.    " 
().  Reduce  -fV  ^^1  a  bushel  to  the  fraction  of  a  gill. 

7.  Reduce  -f  of  a  yd.  to  the  fraction  of  a  nail. 

8.  Reduce  f  of  a  cord  to  the  fraction  of  a  cu,  ft. 

9.  Reduce  Vg-o*  a  gallon  to  the  fraction  of  a  pint. 
10.  Reduce  I  of  a  mile  to  the  fraction  of  a  furlong. 

190.  ExAMPi.K  1. — Reduce  -^qr.  to  the  fraction  of  ,^,^,  ,edu^'« 

a  fcthilling.  fraction  fron»  o. 

OPERATION. 

\^r.=-^i.^4=id.=^l  2=tV^-.  Ana. 
The  fraction   is   divided,  it  will  be  seen,  by  the 
numbers  necessary  to  find  the  required  denomination. 

2.  Reduce  --r'gr.  to  the  fraction  of  a  pound  Apo- 
tbecarics'  weight. 

3.  Reduce  -^^gr.  to  the  fraction  of  a  pound  Troy 
weight.  A71S.  r? 0- 

4.  Reduce  -^f-O-aee.  to  the  fraction  of  an  hour. 
Reduce  ^^  of  a  gill  to  the  fraction  of  a  gallon. 
Reduce  -^  of  a  cubic  foot  to  a  cord. 
Reduce  Vpt-  ^'^  the  fraction  of  a  bushtd. 


lower  to  one  of  k 
higher  name. 


5. 
6. 

7. 

s. 

0. 

l(t. 


Reduce  ^in.  10  the  fraction  of  a  yard. 
Reduce  fd.  to  the  traction  of  a  £. 
Reduce  V'"''    -'^  ^'"^  fraetiofi  of  u  pound. 


100 


VULGAR   TRACTIONS. 


To  reduce  a 
fraction  of  a 
Wgher  denomi- 
nation to  whole 
nnmbert^  of  a 
lower  dc-nomi- 
iiation. 


To  reduce 
whole  numbers 
■of  lower  to  frac- 
tions of  a  high- 
er  donoTiiiixa- 
tion. 


191.  Example  1. — Eeduce-aV  ^f  ^  shilling  to  pence 
and  farthings. 

OPERATION. 

-^S-=-/rXl2=J^=l=4k/.  and 
\-=iqr.x4==^=-2qr.;  hence, -^=4d.  2qr.  Ans. 
We  hero  reduce  the  given  fraction  to  one  of  the  next 
lower  denomination,  by  uialtipl^yiug  with  the  number 
leading  to  sucli  result.  The  improper  fraction  obtained, 
being  reduced  to  a  mixed  number,  its  fractional  part 
was  reduced  to  the  next  lower  denomination  by  mul- 
tiplying with  the  number  necessary  to  such  result. 

Wote. — When  the  improper  fraction  reduced  is  a 
whole  number,  the  result  required  is  obtained. 

2.  Reduce  -j^  of  a  mile  to  furlongs,  chains,  etc. 
o.   Reduce  -i^f-  of  a  yard  to  quarters,  nails,  etc. 

4.  Eeduee  -iV  of  a  month  to  hours,  minutes,  etc. 

5.  Reduce  ^  of  a  bushel  to  pecks,  quarts,  etc. 
f>.  licduce  4-  of  a  circle  to  signs,  degrees,  etc. 

7.  Reduce  -^  of  a  .ton  to  cubic  feet  and  inches. 

8.  Reduce  -^  of  a  hhd.  to  quarts,  pints,  etc. 
9.#lcducc  -^f-  of  a  cord  to'cvibic  feet  and  inches. 

10.   Reduce  -i\-  of  a  eii"cumferenc 
longs  and  rods. 
.,  192.    Ki':.\^JpL-E    ].— Redu<  . 
fraetionof  a  shilliiiij;. 

.   Ol'EkA'il'   .. 

'V/.-f  2^r.=M>4g/'.'  and  Ts.'  (tjQ  which    tht;y  arc   to    be 

reduced")  =48^?''. ;'  hence;  4f  is  fouiid  =^s..  Ans. 
Here,  the  .given   quantity  is  reduced  to  the  lowest 
denominaiioa  possible,  foi*  a  numerator;  and  a  unit 
of  the   hi::-)icr   denomination   is    reduced    to    a   like 
denominat!"n  with  the  ndtiierator,  for  a  denominator. 

2.  Reduc-e  9.5  to  the  fraction  of  a  pound. 
-    3.  Reduce   2pk.  Iqt.  1-iVpt-  to   the  fraction  of  a 
bushel. 

4.  Reduce  2oz.  42dr.  to  the  fraction  of  a  pound. 

5.  Reduce  5  cord  feet,  8eu.  ft.  and  1036-|-in:  to  the 
fraction  of  a  cord. 

6.  Reduce  8ft.  576in.  to  t?he  fraction  of  a  ton. 

7.  Reduce  2fu]-.  25rd.  to  the  fraction  of  a  mile. 

8.  Reduce  15°  40'  to  the  fraction  of  a  sign.  / 

9.  Reduce  oOgal.  oqt.    Ipt.   to    the   fraction   of  a 
hogshead. 

10.  Reduce  5ch.  8rd.  20H.  to  the  fraction  of  a  furlong. 


•e   to  miles,  fur* 
:.^qr.    to    the 


VULGAR    FRACrJONS.  '  101 


ADDITION  OF  FRACTIONS. 

193.    Kx AMPLE  1.— Add  -,V  iU'd  -iV  to,^■ethol^  Tbo  additicm  »f 

■f^.  Ajis. 

OPEHATIOJf.: 

The  denominators  boinjr  ofitho  8a#io  order,  the 
numerators  are  added  and  written  over  the  common 
denominator. 

j!^ote. — When  the  fractions  ai>e  compound,  complex, 
or  mixed  numbers,  reduce  to  simple  fraictioiis. 

2.  Add  too;ether  -,V,  n^s-,  -rV  i^»<l  1^- 

3.  Add  toivcther  -^,-  -^u",  i}  and  ^t. 

4.  Add  together  -,V>  nV  and  -jV-  ^«-5.^  -H-=1tV- 

5.  Add  together  -^,  if,  if  and  ff.        ' 

6.  Add  together  f  and  i. 

OPERATIONS  ,       ,.; 

f=-|4;  |=if-i-2  for  the  least  cpmuionj^ileiiomi^fition, 

2  4    ailU    2  4)- 2  4-     *^"'>-  , 

•    7.  Add  together  |,  -,V  and  -j^g-. 
8.  Add  together  «,  f,  -,V  and  -^V-     ? 
0.  Add  together  3;]  and  51.  Ans.  9|i. 

Note. — In  adding  mixed  nun;jjbei*8,  ifeis  oftQp'more 
convenient  to  add  the  whole  numb^rfjflby  thoEpiaelves, 
and  then  the  fractions. 

10.  Add  together  6s,  7}  and  4^. 

11.  Add  together  41,  5s  and  6|. 

12.  Add  together  2^,  J,  f  and  i.  f 

13.  Add  together  1  of  i,  I  of  f  of  1. 

14.  Add  together  ?i   I  and  -^  off.-  .4n*-.  2if  H- 

^*' 

15.  Add  together  !_  f  of  i  and  f . 

"*5, 

16.  Add  together  is.  and  id.  , 

OPERATION. 


■.2^d.  Afis. 


17.  Add  together  fib.  is  ?o  and  B  - 

18.  Add  together  Jgal.  iqt..  and  fpt.    ilns.  S-iftV-pt- 


lOL'  ,  VV^iG-^U    FRACTIONS 


SL'KTKACTION  OF  FRAOTIOXS. 

The  subtrao-         li}t.  ExImple  1. — From  :i  take  ',. 

tion  of  frsc-  » 

tiona.  OPERATION. 

:|— ,l=f.  A71S. 
^      ^  Here,  the  denominators  being  of  tlie  same  order, 

a  common  dT- the  ditfercnce  of  thfe  numerators  is  found,  and  placed 
nom-nator.        Over  the  cottimon  denominator. 

2.  Fromi.4f  take  -aV. 

3.  From  -ii  take  ii. 

N'ote. — When  tlie  denominators  are  not  alike, 
reduce  them  to  a  common  denom'inator,  and  then, 
subtract  as  above. 

4.  From-S  take  -V. 

i  '  OPERATION. 

|=|- and  2=f=o-  A)is. 

5.  .From  f  take  -f . 
6..  From,  I- take -f. 

'    7i  From^'?^-  take  i. 

8,  From  ^  take  -H. 

9.  From  i-  off  take  i  off.  A71S.  -^. 

10.  From  9i  take  If. 

Wheu  Uiere  .we     iVb^6\— Whenever  there  are  compound,  complex  or 

comppund  or    mixed  numhers,  brini?  them  to  their  simplest  forms, 
complex  trac-  ■,     y  i 

tioBSror  mixed  and  then  subtract, 
nwnbars. 

11.  From  23-iV  take  15-,V- 

12.  From  120  take  40f . 

N^ote. — Change  the  whole  number  to  5ths. 

13.  From  265  tak«  52f . 


VULGAR    FRACTIONS.  103 


Ml  SCELL  ANKOUSEX  AM  P  LKS. 

t 

IftS.  Example  1. — If  lib.  of  beef  in  worth  S.J  eentR, 
what  is  the  value  of  9. libs.  ?  ■  ;« 

'1.  Bought  a  firkin  of  butter  eontiihiin  j  40] U».,  and 
was  charged  '111  per  lb. :  what  did  it  ainotmt  to  ? 

3.  When  barle}"^  is  worth  $1|  ft  biishol,  what  will 
24>V  bushels  cost?  ^  ' 

4.  If  25bbl.  of  fldnr  coist  $12'^].  what  will  O^bbl. 
cost  ? 

5.  When  corn  is  soiling  for  ','  dollar  a  bushel,  what 
must  be  paid  for  250 J  bushels  y 

6.  If  you  Avish  to  divide  $130  among  soiue  laborers, 
so  that  7  of  ther.i  should  have  each  f  as  rhuch  as  each 
of  the  other  3,  what  would  you  give?      ' 

7.  lf2Jyd.  of  cloth  will  pay  for  33;}lb  of  caudloS, 
what  quantity  of  cloth  Avill  pay  for  7]  times  S3  Jib.  of 
candles  ?  '* 

8.  If  for  -iV  of  a  bushel  of  pO»tatoes  you  pay  42 
cents,  how  many  could  be  had  for  SfS-j*^? 

Am.  lOibush. 
0.  How  niau}-  cubic  feet  are  there  in  ~^x~  of  a.  cord 
of  wood,  and  what  is  it  worth  at  2">2  ))cr  cord? 

Am.  104Aft.;  U\. 
10.  32  is  -§■  of  how  many  times  \  of  12  ? 

Ans.  9  times. 
U.  Sold  73^bush.  corn  for  §04|-.J :  what  Avill  be  the 
amount,  at  the  same  rate  of  G4bufih.  ?     Ans.  $50. 

12.  28  is  i{  of  how  many  times  8  ? 

13.  How  many  cubic  feet  in  a  box  that  is  Onft.  long, 
r>f  ft.  wide,  and  3-J^ft.  deep  ? 

OPKRATION. 

t)  a  X  5 4  X  3i=  V  X  V X  -V»= ^^3*6^=  ^i^=  11 7ii.  .l«s. 

14.  In  a  box  that  is  8ift.  long,  4fft.  wide,  and  3*ft. 
dc(^p,  how  -many  cubic  feet  are  there  ? 

15.  Of  the  inhabitants  of  a  towii  in  Alabama,  I  arc 
planters,  f  merchants,  |  students  and  professional 
men,  -J-  mechanics,  and  142  others  variously  engaged  : 
what  iR  the  number?     Ana^.  5040. 


104 


VULGAR   FRACTIONS. 

16.  If  the  cargo  of  a  ship  be  worth  $72,000,  and  if 
f  of  J  of  ■?■  of  the  cargo ^  be  worth  f  of  i  of  -f^  of  the 
ship  :  what  is  the  value  of  the  ship  ? 

17.  In  a  school  in  Georgia,  2  the  schoLars  study 
arithmetic,  ;}  algebra,  -^  geometrj-,  and  the  others,  in 
number  10,  study  engineering:  how  many  scholars 
are  there  ?     Ans.  200. 

18.  The  factors  of,;a  certain  number  are  32i,  15-f 
and  19f :  wh^'^t  is  f  of  f  of  |-  of  the  number  ? 

>^  Ans.  82231-^. 

19.  How  much  cloth  that  is  -f  of  a  yd  wide,  will  it 
take  tp  line  a  cloak,  cpntaining  84  yd.  whjch  is  -H  of  a 
yd.  wide  ?     Ans.  12  f |  yd. 

20.  What  is  f  of  a  barrel  of  flour  worth  at  $6f  per 
barrel  ? 

21.  A  man,' can-  build  33 J  rods  of  wall  in  24 J  days, 
by  laboring  123  hours  per  day :  in  how  many  days  of 
9f-- hours,  will  he  bui]d  1^  times  as  many  I'ods? 

22.  A  gardwi  whose  breadth  is  10  rods,  and  whose 
length  is  If  times  its  breadth,  has  a  wall  3^  feet  thick 
around  it :  what  was  the  cost  of  digging  a  trench  2f 
feet  deep;in  which  to  lay  this  wall,  at  f  cent  per  cubic 
foot  ?    Ans.  $62.941. , 

23.YThe  distance  f^om  the  earth  to  the  sun  is  about 
95,000,000  miles :  in.what  time  would  a  railway  car 
run  that  distance,  at  the  speed  of  372  miles  an  hour, 
allowing  3651  days  in  a  year  ? 

Ans.  288yr.  363d.  13h.  20min. 
24.  What  is  the  value  of  1^3   of  a  day  ? 

Ans.  16h.  36min.  55-iV- 


DECIMAL    FRACTIONS.  'lOA 


DECIMAL    FEACTIONS. 

190t  A  Decimal  Fraction  is  one  that  has  its  denorai-  ^'.^^  ^^*"°;r,'''' 
nator  written  in  tQnths,  or  similar  classes  of  num- fined.""  ' 
bers. 

197.  This  denominator  is  not  usually  expressed,  How  the  de- 
bufit'is  known  from  the  annexation  to  its  figure  1  o/knwn^*'"^  '" 
as  many  ciphers  as  the  decimal  demands. 

198.  A  decimal  fraction  is  distinguished  from  a  How  a  dooimai 
whole  number  by  a  dot,  known  as  the  decimal  i>o<'^^,  [[n°ui>ho.'i. ''"''' 
placed  at  the  left  of  the  decimal ;  the  first  figure  at 

the  right  of  the  point  being  tenths,  the  second  hun- 
dredths, the  third"  thousandths,  etc. ;   thus  .b=-^tr', 

NUMERATION    TABLE    OF   DECIMALS. 

199.  In  the  decimal  table,  w^hich  is  read  from  left  How  the  depi- 
to  right,  it  will  be  seen— as  tenth  denotes— that  the  ;-;\^|^'^u|^"<!;f;;;t 
numbers  decrease  in  value.  In  the  table  of  whole  whole munhors. 
numbers  it  is  remembered  that  the  value  increases 

from  right  to  left.  Similar  names  to  designate  values 
are  used,  always  placing,*or  supposing  to  bo  placed, 
the  integer  1  where  the  unit  belongs,  with  its  distin- 
guishing or  separating  point. 


cnTenths. 
Hundredths. 
Thousandths. 
Ten  thousandt 
Hundred  thoui 
Millionths. 
Ten  milUonthg 

;  5  tenths. 

Decini.<il  num^ 
ration  table. 

.6  7 

67  hundredths. 

^ 

.0  9  9 

99  thousandths. 

. 

.45677 

45o7  ten  thousandths. 

.02345 

2845  hundred  thousandths. 

.004789 

47S9  millionths. 

0  0  3  4  5  6  7 

34567  ten  millionths: 

8* 

106- 


DECIMAL    FRACTIONS. 


Why  decimal 
fractions  are 
written    'Ji  this 
way. 


Y.alue  not  alter- 
ed by  annexing 
ciphers. 

When  its  value 
is  decreased.    . 


Whole  nuTYi- 
bers  and    deci 
nials  written  to 
.gether. 


'1  able  of  whole 
numbers  and 
decimals  to- 
getlKT.' 


2€'0.  Decimal  Fractions  are  thus  .written  to  avoid 
the  confusion  that  arises  from  the  denominators, 
when  expressed.  These  denominators  are  ahvays  un- 
derstood; thus,  .67  may  be  read  -jV  "i^^'^  ihi,  ov  -^^, 
which  is  the  equivalent  expression. 

201.  The  value  of  a  decimal  is  not  altered  by  the 
annexation  of  a  cipher  or  ciphers  ;.  thus,  -^=-^A- 
=  to-^-o ',  equivaleutly,  these  are  .6=. 00=  .600.  ' 

202.  The  value  of  a  decimal  is  decreased  to  tV  of 
its  first  value  when  a  cijiher  is  prefixed  to  it,  since  it 
removes  the  figure  one  place  from  the  decimal  point ; 
thus,  .(i=TV,  b'lt  .06=t1o-,  which  is  but-iV  of  nV- 

,2®Ji  Whole  numbers  ajid  de'cimals  cun  be  written 
together  and  easily  read  when  the  decimal  point  is 
placed  between. 

.  201.  TABLE. 


=1      .    ^  ^3 

•  5           -^ 

■"3   q!         o!   2     - 

"2    'T^    O    C, 

O'  +J    m 

•  73   O        r^--^ 

"+-  ;^  'sH  p! 

m   C    m  ri-^  Ti   <c 

rS  rS   a>         «:■   2    "    K 

T3    q    S'^^S'^  — 

CO  o  'V(   o)  =*-(  S  o 

•  ©  tj  o  o  "tf  r:::  o  -1.^ 

c=   i-  c  ^  o  S   1^ 

» 

5;-.5^j-Cr;rtfl 

Billioi 

Hund 

Tens 

Hund 

Tens  ( 

Thouf 

Hund 

Tons. 

9  8  7,6  5  4,8 

2 

1 

0234  5  6789 

AVhole  numbers.  Decimals. 

205.  When  a  mixed  numb^  has  the  decimal  pointy 
it  must  be  as  a  fraction  of  a  unit  of  the  order  repre- 
sented by  the  preceding  decimal  figure ;  thus,  in  th« 
mixed  number  .2^,  the  \  is  half  of  a  tenth  ;  in  .22^,  it 
is  J  of  a  hundredth. 


"Notation  C'f  de- 
f^imal  frartions. 


NOTATION   OF   DECIMAL   FRACTIONS. 

Write  in  figures  : 

Example  1. — Thirty-five  hundredths.  .  Ans.  .35. 

2.  Fifteen  thousandth. 

3.  Fifty -five  tenths  of  milliontbs. 

NoU. — It  saves  confusion  in  writing  whole  numbers 
and  decimals  together,  to  place  the  word  decimal 
before  the  fraction  ;  also,  when  a  decimal  only  is  ex- 
pressed. 


DECIMAL    FRACTIONS. 


107 


4.  Five  hundred  and  decimal  two  thoii:3andLhs. 

5.  Foiu'-  tlioiit^and  and  decimal  six  thousandths. 

6.  Six  hundred  and  fifty  and' decimal  throe  thou- 
sandths. 

7.  Eighty-five    and    decimal    seventy-six    thou- 
sandths. 

8.  One   hundred   and   twenty   and   decimal  fifty 
thousandths. 

9.  Two  hundred  and  sixty  and  decimal  fifty-six 
tenths. 

10.  Three  hundred  and  decimal  six  hundredths. 

11.  l'>c(-imal  three  thousandths. 

12.  Dccinial  t^Vo  hundredths. 

206.  Addition,  subtraction,  multiplication  and  di-  How  aiiditk-u. 
vision  of  decimals  ate  performed  precisely  as  whole  formed/"", 
numbers. 
Example  1. 

2.  0.56721 

3.42501 


Add  toge 

ther    4.56 

86.902 

Sum, 

41.462 

Proof, 

41.462 

8. 

805.78402 

270.23521" 

6543.12678 

5005.67812 

2.16105 

13182.98548 

'2-22  12211 

12.90222 
12.09222 

534.21005 

60.51054 

6.05678 

.00525 

325.60789 


.  Remariv. — Bern embev  always  to  place  tenths  under 
tenths,  hundredths  under  hundredths,  thousandths 
under  thousandths,  etc.,  and  the  point  in  the  sum 
total  directly  uiider  the  numbers  of  the  work  given, 
to  be  done. 


5.  Add  2.40,  545.2,  676.'0006  and  3.57580. 

6.  Add  1765.5,  52.4301,  6.0065  and  .9536. 

7.  Add  5544,  39.7,  678.0212,  .00007  and  31.5.' 

8.  Add  .70l»,  .560,  .4809,  .395,  .56789  and  .3366. 

9.  Add  29.59,  5000.004^  200.03  and  56.547. 

10.  Add  decimal  three  hundred  and   three  thou- 
sandths,  thirteen,   decimal   forty   thousandths,   two 


DECIMAL    FRACTIONS. 

hundred  and  decimal  two  thousandths,  thirty -five 
milhons  and  decimal  fourmillionth8,decim^al  thirteen 
thousandths.  .Ans.  35000213.358004. 


SUBTPtACTION. 

OPERATION. 


207.  Example  1. 
From  9.6542 

Take  8.5431 


Rem. 
Proof, 


1.1111 


9.6542 


36.05946 

28.75437 

7.30509 


From    54.367890 
Take    39.43^07565 


69.057546 
33.556327 


Note. — In  examples  like  the  3d,  place  ciphers  over 
the  figures  in  the  svibtrahend  that  have  none,  and 
then  say  5  from  10,  5,  and  carrying  1  for  what  was 
borrowed,  say  7  from  10,  3. 

4.  From  965.43445  take  45.395426. 

5.  From  874.32154  take  3.4506077. 

6.  From  .895456  take"  .000543670. 

7.  From  1.50050000  take  .60051789. 

8.  From  23.0567  take  22.14675. 

9.  From  67469.  take  .56543. 

10.  From  fifty -four  millions  take  decimal  fifty-four 
millionths. 

Note. — Let  the  pupil  be  careful  to  jilace  the  separat- 
ing points  one  under  the  other. 


DECIMAL    FRACTIONS. 


109 


MULTIPLICATION. 


208. 


Example  1.— Multiply  .30  by  .30. 

OPERATION. 

.30 


.30 

210 
108 

.1290 


Eemauk. — In  the  mul-  „.^  ,        , 

,.    ,.       ,.  pj      .        ,       ,     ^\h.'^t  numbers* 

tipUCatlOll  OI,aeCUnalS  Ob-  in  result  to  Vo 

the  1st  and  P'^'"'^^^  ''"'• 


serve,  as  in 

2d  examples,  to  point  oif 
as  many  Iri^ures  tor  deci- 
mals in  the  product  as 
in  both  multiplicand  and 
multiplier  are  of  this  kind. 

Multiply  50.425 

By         ,      2.5 


28212500 
112850 


141.0025 

Note. — The  ciphers  annexed  arc  merely  foV  conve- 
nience. The  value  of  the  sum  would  be  the  same  if 
the  multiplier  had  been  placed  at  the  extreme  right. 
Such  is  the  preferable  form ;  thus,  50.425 

2.5 


8.  Multiply 
By 


.7284 
.00023 

21852- 
14508 


282125 
112850 

4. 

141.0025 

.5082 
.30 

34092 
17040 

.204552 

.000107532 
5.M\iltiply  507  by  3.24. 

6.  Multiply  .435  by  345. 

7.  Multiply  5.95  by  03.32. 

8.  Multiply  .2350  by  .3453. 

9.  Multiply  05.30  by  234.5. 
10.  Multiply  .0000  by  .X)005. 

Note. — When  there  are  not  as  maDy  decimal  figures 


110 


DECIMAL    FRACTIONS. 


wherf-a.defi-_  in  tho  procliict  US  there  are  decimal  places  in  mul- 
inafllA-ai-es* ex-^^P^^*^^' ''"^^  mulliplickncl,  pliiCG  ag  many  ciphers  at  the 
ists  ufpioduct.  left  as  the  deficiency  requires. 


11. 


Multiply     .15 
By. '  .05 


.0075 


To  multiply  by 
10. 100,  etc. ' 


12.  Multiply  .6250  by  .08. 

13.  Multiply  .5945  by  009. 
To  miiltiply  by  10,   100,  etc.,  move   the   decimal 

point  ag  many  places  toward  the  right  as  there  are 
ciphers  in  the  multiplier.  This  makes  each  figure  in 
the '.multiplicand  10  times  what  it  previously  wa?*. 
Hence,  the  result  is  ten  times  as  great  as  the  multi- 
plicand. • , 

14.  Multiply  .0467  by  100.  Ans.  4.67. 

15.  Multiply  .00454  bv  1000. 

16.  Multiply  8.764  by ^300.   '  Ans.  2G29.2.' 

17.  Multiply  .4)004  by  5600.   ' 

18.  Multiply  3.253  by  2.900.      . 


DIVISION. 


The  number  of 
decimal  places 
so  bo  in  quo- 


209.  Example  1.— Divide  .845  by  .18. 


OPERATION. 

.13). 845(6.5  Am. 

78 

05 
65 


Note. — There  must  be  a* 
many  decimal  places  in  the 
quotient  as  the  decimal 
places  in  the  dividend  ex- 
ceed those  in  the  divisor. 
The  excess  in  this  example 
is  1.     . 


2.4).1248(.052. 
120 


48 
48 


The  excess  here  is 


3.  Divide  2.3462  by  2.11. 

4.  Divide  94.0056  by  .0.8.    . 

5.  Divide  .17638  by  .369. 

6.  Divide  61064.14  by  .4506. 


DECIMAL   FRACTIONS.  Ill 

210.  When  there  arc  moredecimal  places  in  the  divisor  Wiien  u.,;  ■  .« 
than  in  the  dividend,  annex,  ciphers  to  make  the  work  clmainn^aiTi- 
practicable.  ''"r 

7.  Dividc*1941.855  by  .7846.  Ans.  2.475. 

8.  Divide  26043.16  by  .3527. 

211.  Should  the  number  of -figures  iii  the  quotient  be  When  the  qm- 
less  than  the  excess  of  decimal  places  in  the  dividend  over  [g!;"'  f^»mheru 
those  of  the  divisor,  prefix  to  the  quotient  tlie  number  of 

<  iphers  which  the  deficiency  requires. 

9.  Divide  .08750  by  1.1.  Ans.  .0700. 

10.  Divide  .000702  by  .12,  .l»,.9.  .0006.  _ 

212.  When  it  is  desirable  to  extend  the  division  beyond  ^-:.  „  ,;_j;i.,. 
the  given  numbers,  ciphers  can  be  annexed.    When  it  is  not  *'■'-  auuexeii. 
wished  to  go  beyond  a  certain  figure,  or  when  it  can  be 
continued  beyond  the  given  niimber,  we  annex  the  sign4- ; 

thus,  4  divided  by  .7=''.571+.  •    . 

11.  Divide  .306656  by  .5  to  thd  extent  of  Ihousandths. 

•    OPERATION. 

5\306656 


.611+ 


12.  Divide  .5876i\  by  .6  to  hundredths.  inaifr.«lr..u»l 
Decimal    fractfons  arc   divided   by   10,   100,  1000,  byio,  etc. 

moving  the  point  as  many  "places  toward  the  left  as  there 
arc  ciphers  in  the  divisor.  By  this,  each  figure  in  the 
dividend  becomes  only  ^g  as  much  as  it  previously  was,  and 
«n  the  result  is  correspondent;  thus,  567.5^t'0=56.75.- 

13.  Divide  7846.987  by  1000.  Am.  7.846987.- 

14.  Divide  .9;)67  by  300. 
.    15.  Divide  .7986  by  500. 

15.  Divide  .54789  by  6000. 


KBDUCTION. 

213.  Example  1.— Reduce  i  to  a  decimal  iVaotion.       I^Tt^n'^^vr: 
Remark. — The  value   of    a  fraction  is    the   quotient iin'-U^" 
'(rising  from  the  division  of  its  numerator  by  the  denom- 
inator.    This  value  remains  unchanged,  though  a  cipher 
or  ciphers  be  annexed.     Thus,  to  reduce  if  wo  arin<'x,  say. 
three  ciphers,  and  divide  by  its  denominator  4. 

OPERATION.  I       Here,   aa  mimy   decimals 

.4). 3000  are  pointed  off  ill  the  result 

—  i  as  thQ  decimal  places  in  the 

.750  j  dividend  exceed  those  inthe 

divisor. 


112 


DECIMAL    FRACTIONS 


2.  Keduce  -i*,-  to  a  decimal. 

3.  Reduce  -3^  to  a  decimal. 

4.  Reduce  -^q-  to  a  decimal. 

5.  Reduce  -^  to  a  decimal. 

6.  Reduce  ^  to  a  decimal. 

7.  Reduce  to  a  decimal  -of  three  figures  f,  -f-,  i,  -^. 

8.  Reduce  to  a  decimal  of  five  figures  -^,  -^^,  -^f. 
changed *to  a        214.  A  decimal  fraction  is  changed  to  a  vulgar  fraction, 
vulgar  fraction. by  vFriting  beneath  it  its  proper  denominator;  thus,  .5= 


.05= 


0,^5=— B^i- 
.VOO 10  0  0- 


9.  Reduce  .15. 

10.  Reduce  .350. 

11.  Reduce  3.5. 

12.  Reduce  5.65. . 

13.  Reduce  6d.  and  3qr.  to  the  decimal  of  a  shilling. 

.  3qr.=  td.==.75;  thus,  6d.  3qr.=6.75.  Am. 

14.  Reduce  10s.  5d.  Iqr,  to  the  decimal  of  a  pound. 

OPERATION. 

4)1.00qr. 


12)5.2500d. 


20)10.437503. 


lqr.=  .25d.;  5.25d.=.43758. 
10.43'75s,=£521875. 


£521875.  Ans. 

Explanatory  re-     REMARK.: — Ciphers  *are  annexed  to  the  lowest  denomi- 

ti(5*n*^  *"*  opera-  nation,  and  its  division  performed  by  the  number  of  that 

denomitfation  which    it  takes   to  make   one  of  the  next 

higher.     The  quotient  is  annexed  as  a  decimal  to  the  next 

higher  number  to  be  reduced,  and  so  on  to  the  result. 

15.  Reduce  6oz.  18dwt.  15gr.  to  the  decimal  of  a  pound 
•                 Troy  weight.  Ans.  .5776041661b. 

16.  Reduce  65    23    29    2gr,  to  the  decimal  of.a  pound. 

17.  Reduce  5yd.  2ft.  6in.  to  the  decimal  of  a  rod. 

OPERATION. 

12      e.Oin.    ■ 


11 


2.500ft. 

5.8333+yd. 
9 


11.6666+half  yd. 


1.0606+rod.  Ans. 


DECIMAL    FRACTIONS. 


11;; 


18.  Reduce  43a.  or.  25rd.  20yd.  Gft.  SOin.  to  the  deci- 
mal of  a  sq.  mile. 

19.  Jleduce  25ft.  lG75iu.  to  the  decimal  of  a  cord. 
•  20.  lleduce  3qr.  ona.  to  the  decimal  of  a  yard. 

21.  Keduee  2pk.  6(]t.  Ipt.  to  the  decimal  of  a  bushel. 

22.  Reduce  £421875  to  shillings,  pence  and  farthiugs. 

Ans.  8s.  5d.  Iqr. 

OPERATiaX.  EXl'LAN'ATIOX. 

£421875  ^  The  first    multiplication  Expianatfon 

20s.       ,  changes  the  decimal  of  the''""  "'"'■ 

—  '  pound  to  the  shillings  and 

8.437500  its   decimal  of  a   shilling; 

12d.  ;  the    second    multiplication 

changes  the  decimal  of  the 

r^.J.VJO  shillings  to  pence,  and  so  on. 

4qr.  As  many  places  are  pointed 

off  in  each  product  for  deci- 

1.00  mals  as  there  are  decimals 

1  in  the  example.    The  answer 

I.-:  the  several  deuominations  On  the  left  of  the^ecimal  point. 

N^ote. — The  ciphers  omitted  in  the  multiplication  are  to 
l)c  counted  with  the  other  figures,  to  determine  the  posi- 
tion of  the  point.     They  were  omitted  for  convenience. 

23.  It^duce  .9375  of  a  gallon  to  qt.  pt.  and  gi. 

Ans.  8qt.  Ipt.  2gi. 

24.  Rediice  .7694  of  an  acre  to  rd.,  etc. 

25.  Reduce  .84  of  a  lunar  month  to  wk.,  etc. 

Ans.  3w.  2da.  12h.  28rain.  48.'5ec, 
0 


114  MISCELLANEOUS   EXAMPLES. 


MISCELLANEOUS  EXAMPLES. 

215.  p]xAMPLK  1. — A  grocer  sold  16  bushels  of  potatoer. 
at  Q.iS)  per  bushel ;  o  baga-  of  coffee  at  315.G:i5  per  bag; 
and  14it  barrels  of  flour  at  $G.50  per  barrel :  what  was  the. 
amount 'r""  ^ 

2,  A  liousekeeper  purchased  561bs.  of  sugar  at  12.5 
per  lb.;  151b.  of  currants  at  18.75  per  lb.;  51b.  of 
almonds  at  16.33  per  lb.,  and  61b.  of  starch  at  6  25  per 
lb,':  what  was  amoitnt  of  purchase? 

3.-  Bought  a  keg  of  butter,  containing  961b.,  at  21.5 
■per  lb.,  but  sold  one  half  pf  it  for  25  ceats  per  lb. ;  what 
was  paid  for  the  whole  and  received  for  the  half? 

4.  A  mcrcliant  sold  -V'y.'lyd.  of  cloth  at  W-Od  per  yd. ; 
what  was  his  gain,  the  cost  to  him  being  5.555  per  yd.  ? 

5.  An  advc^iture  has  gained  for  6  pei'sous  ^175.035  : 
what  is  the  share  of  each  ? 

G.  A  merchant  bought  14Jhhd.  of  wine  at  $75,333  pe* 
hhd  ,  and  sold  them  at  public  vendue,  when  they  brou^tt 
only  .^00.50  a  hhd.,  exclusive  of  commissions  and  c^her 
expenses  -t-SS  :  what  was  bis  loss  ? 

7.  Bought  19  barrels  of  flour  for  095.125:  w^at  was 
the  cost  of  a  single  barrel? 

8.  Imported  65  bags  of  coffee  at  a  cost  of  i'996.555: 
what  was  the  cost  of  a  bag  ? 

9.  A  trader  bought  iJ,  barrels  of  pugar  a/ $18. 7 5  per 
bbl  :  wiiat  was  the  cost  of  each  barrel  ?        / 

10.  A  bag  cf  cotton,  weighing  ooOIb.,  ^as  shipped  to 
Liverpool  and  sold  at  '6'6M  per  lb.,  Atnerican  money. 
Deducting  various  expenses,  $0,125,  wb^t  did  it  net  the 
shipper?  / 

11.  A  factor  sold  26cwt.  3qr.  cf  rirfc,  in  Liverpool,  for 
£2  IGs.  per  cwt.  :  what  was  the  amoutit  ? 

12.  Bought  30bush.  opk  of  wheat  at  $1,375  per  bushel : 
what  .was  amount  of  purchase? 

IB.  Bought  2  bancls  of  flour  at  §6}  jer  bbl  ;  4  bushels 
cf  corn  at"62l  cents  per  bushel;  6ib.  of  coffee  at  16j 
cents  per  lb.;  101b.  of  sugar- at  61  conta  per  lb.,  and  6Ib. 
or  butter  at  18 J  cents  per  lb. :  what  did  they  avcuanb  to  ? 

A'Oi'i'. — Change  fractious  to  decimals. 


MISCELLANEOUS    KXAMPLE8.  liO 

14.  WIi:it  would  be  the  cost,  of  building  37ni.  Clur. 
22rd.  of  railroad  at  §8355.(521  per  mile  ? 

15.  A  merchant  bought  20hhd,  of  tobricco  at  $57J  per 
hhd.,  and  sold  them  for  S15o5i  :  what  was  his  proUt? 

16.  Wliat  is  the  value  of  .575cwt.  of  coal  at  £'d  Gs.  Gd. 
2qr.  per  ton  ? 

17.  What  is  the  value  of  10  bales  of  Sea  Island  cotton, 
the  average  of  the  bnles  being  3201b.,  at  4;j|  cents 
per  lb. 

18.  What  must  be  paid  for  52  weeks*  board,  at  the  rate 
•^Hch  week  of  !^4.o7J  ?  - 

19.  Ao  2ri  cents  a  bushel,  J^hat  number  of  bushels  can 
be  had  for  $200^  ? 

20.  A  piece  of  land  is  39.5  rods  long  and  25. 5G  roda 
wide  :  what  will,  it  cost  to  wall  it  at  G2t}  ccnta  per  rod  ? 


11(3  CIRCULATING    DECIMALS. 


OIRCULATI A (.i    lJi.CiMA L>. 

•inr/de'fined^'  .  ^^^^'  ^'^  Circnlnlifuf  Derm  , I  is  simply  a  decinnil  frac- 
tion, repeating  invariably  the  same  figure  once  or  many 
times. 

When  a  single      217.  When  confitied   to  a  sirisrle  fijrure  it  h  cnllcd  a 

'.poUnd:  .,  ,,  „.,_  ,^         ii-*  i 

,.„  jj  ^.fJJy^,    Single  repotend,  a.s  .60S,  etc.}  beyond  this,  a  compound 

:   imd:     .       one,  as  010101,  etc.;  but  when  combined  with  other  . deci- 

whcm  niixed:   mals  preceding  its  cxprcyKion,  a  mixed  repetend,  as  .8333. 

hen  ft  finite   ^^^  figure  ov  figures  preceding  the  repetend   is-  r;ill"d  the 

"•I-  finite  p:irt  of  the  cx])ressioH,  as  S. 

ii.ivv  the  repe-      !21S.  A  Tepetoyd  is  obtained  by  reducing  a  vulj^ai  iiac- 

imdishaiL       fJQj^  jq  ,,  decimal  M'hen  there  is  any.  prime  factor,  except  2 

and  5,  in  the  denomiBator,  and  not  numerator,  as  ^=^.HoS, 

etc. ;  4=''"^5''i>  etc. 

The  way  trf  !il».  To  avoid  repetition  in  a  simple  number;  the  same 

"^"-  is  written  once,  with  a  point  above  it  j  thu:^,  4=.,'). ;  but 

in  a  compound,  the  first  and  last  figufcs  are  thus  desig- 
nated. -^3^^=  1 08. 

wiiat  is  a  per-     220.   When  the  number  of  figures  in  the  n!p9tend  is 

lect  repotend.  ^^g  less  than  the  number  of  units  in  the  denominator  of 
the  equivalent  vulgar  fraction,  it  is  called  a  perfect  repV 
tend;  thus,  -f  gives  the  perfect  repetend,  .14'J857. 

^■'»fVinixud  ^*^^'  'I'o  fi?id  the  value  of  a  mixed  repetend,  iii?certain 
ietend.  the  valuc  of  the  finite  part  and  cf  the  repetend  separately, 

and  add  the  re?uUs. 

When  one-  222.  KxAMPLj!  1. — What  js  the  value  of  .270? 

ninth  IS  re-  ■  . 

duc-ed  to  n  dcci-  .275=27+-4-5=-Hi+-r&  f=iU=/aV-  ^ns. 

Note. — When  a  decimal  begins  to  repeat  at  the  third 
place,  the  two' first  figures  will  be  so  many  hundredths, 
■  and  the  repeating  figure  so  many  ninths  of  another  hun- 
dred: 

2.  What  is  the  value  of  .345  ? 

3.  What  islhe  value  of  .67123  t 

4.  Change  .4444,  etc.,  to  a  vulgar  fraction.       Ans.  \. 

5.  Change  .9999,  etc.,  to  a  vulgar  fraction. 

Xote. — When  \  is  reduced  to  a  decimal  it  produces  a 


CONTINUED   FilACTIONii.  117 

quotient  of\llll,  etc.,  that  is  the  repetend  1 ;  4  is  the^^''V'"  ».  "^eci- 

1  '  '  .  '  _  '     *  niiil  begins  to 

value  of  the  repetend  1  ;  the  value  of  833,  etc.,  or  the''<"peat  ut  tho 

'■  '  ■  }  7  third  place. 

repetend  3,  must  ho  three  times  as  much,  or   -f;  4,  f;  5, 

■^;  andi),  A=il. 

6.  Chance  .5833,  etc.,  to  a  vuli;ar  fraction. 

Here,  the  figure  5=-i\  and  the  remaining  part  of  the 
fraction  is  -f  of  -jV)  that  is  -v,^=-/u-  J  add  these,  A==io"~l"'/(r 
=iu-"=T^-  ^"^-     ft"  changed  back  will  give  .5833. 

7.  Change  .3744,  etc.,  to  a  vulgar-fraction. 

8.  Change  .40355,  etc.,  to  a  vulgar  fraction. 

9.  Change  .863630,  etc.,  to  a  vulgar  fraction. 

When  -gV  is  changed  to  a  decimal  it  produces  .010101,  ^^^'":"  (^"r"^"®- 
etc.     The  decimal  .030303,  etc.,  is  three  times  as  much,  cLianL'ed"to  a 
and  is -,3^j.— -gijj     The  decimal    363636,  etc.,- is  thirty-six^^^^'"''''- 
times  as  much,  and  is  -f  9=-i*r.  ' 

10.  Change  .47647  to  a  vul.Q;ar  fraction. 

11.  Change  .•24' to  a  vulgar  fraction. 

12.  Change  .42  to  a  vulgar  fraction. 

13.  Change  .72  to  a  vulgar  fraction.         Ans.  -{-5=  iV 

14.  Cli;^!'>.<'  On:^  to  a  vul^-ar  -i'mction.    .!;/«.  ~-^»-=-T-i-T. 


COXTIXUKO  rKACTIONS. 

S'J3.  A   Continual  Fracfian  is  one  which  has  for  its  ^  contiiniod 
numerator  a  unit,  and  for  its  denominator  a  whole  number  fined, 
plus  a  fraction,  and  si)  on  ;  thus, 

'  '  >'+ete.,  is  a  continued  frartion. 

S24.  The  investiiration  of  the  nature  and  properties  of  its  investi^n^ 
this  fractional  formula  belongs  to  the  study  of  Algebra,  ^XiRsTAige^ 
and,  hence,  wo  give  simply  a  passing  reference  to  the  sul>  bra.  ;^ 

ject.  « 

Example. — Reduce  -J^  to  a  continued  fraction. 

Ol'EKATIO.V. 

i^= '-+,%-  and  A=i+| .  ■  hence,  il=  ^^^^_  ■ 

Again,  1=++^.    j^^^^^^    ii=i.x_^. 


118  DUODECIMALS. 

EXPLANATION, 

We  here  dlvkle  botli  terms  of  -ff  by  17^  the  numerator 

of  the  fraction,  and  get  i_,.^4..  then  dividing  both  terms 

of  A  by  9,  Ihe  numerator,  we  get  i.|_A .  substituting  then 

this  value  of  -^  in  ,the  expression  l\ ^L 

wehavei^=J_}_j. 

'  '  t-f,  etc. 

225.  In  any  continued  fraction,  ^-.^1 

''  >4-i,  the  several 
aimple  fractions  arc  known  as  integral,  fractiovis,  because 
'  the  denominators  are  integers  j  thus,  ^  in  the  above  is  the 
first  integral  fraction,  \.  is  the  second,  -f  is  the  third,  and 
^,is  the  fourth.  Sometimes  we  call  ^  the  first  approxi- 
mating or  converging  fraction;  ^^_j.  ^^^  second,  and  so  on. 

DUODECIMALS. 

Duod-'.tiimtis        226.  DuodecHiials  are  concrete*  fractions,  and  are  chiefly 

^^' '  used  in  the  measurement  of  surfaces  and  solids. 

How  changed. ^    22^.  They  3X0  added,  subtracted  and  divided,  as  other 
compound  numbers ;  but,  in  certain  cases,  they  are  multi- 
plied differently. 
How  t)iey  do-      228.  Duodecimals  decrease  uniformly  from  the  highest 
crease.  ^^  ^^^  luwest  denomination,  by  the  constant  divisor,  12. 

The  measures.     220.  The  measures  used  lor  their  change  are  the  inch 

or  prime,  the  second  and  the  thirds 
The  measures      2 SO.  A  foot  divided  into   12  equal  parts,  is,   ia  each 
into  twoluhH.     division,  called  an  inch  or  prime;  an  inch,  also  s'.milarly 
divided,  is  called  a  second ;  and  a  second,  a  third ;  thus, 
1  inch  or  prime,  marked  1'=-jV  of  a  foot. 
1  second,  "        l"=-iV  of-iV  oi'  tIt  '^f  a  foot. 

1  third  "       r"=nVof-,Vof-,Vorl788 

"  of  a!  foot,  and  so  on,  for  minuter  divisions,  when  required. 

What  are  jndi-       231.  The  distinguishing  points  are  known  as  indices. 


<?e.9 


ADDITION. 

Example  1.— Add  3ft.  6'  3"  2'"  and  2ft.  1'  10"  11'" 
together. 

*  Concrete  is  a  terra  applied  to  a  particular  object. 


DUODECIMALS. 


IW 


Ol'UnATION. 

12     12 


8. 
2. 


0.  3. 

1.  10. 


12 
III 

2 
11 


Ans.  5ft.  8'     2' 
1 


KX)'t.AN.\TIOK. 


We  first  say  1 1  and  2  are  f  spjanRtioi.  of 


]3;  a,s  this  exceeds 
common  measure,  ll^,  it  ia 
divided  by  12'",  at)d  its  re- 
mainder, 1,  is  put  down, 
and  the  quotient,  J,  carried 
to  10==  II..  We  next  add 
11  and  3,  and  dividitjg  by  12",  put  down  the  remainder 
2,  and  carry  1.  The  other  figures,  being  less  than  1_',  are 
added  like  simple  numbers. 

2.  Add  Sft.  0'  f'  and  Gft.  7'  H"  A"'. 

A  us.  15ft   4'  10"  4'".  '. 
..'..  Add  10ft.  R'  6'"  and  r>ft.  y  y\ 
■!.  Add  Oft.  0'  :V'  and  7ft.  2'  l'\ 
5.  Add  15rt.  C/  V  .-vnd  ()ft.-.  8^H'^ 
().  Add  20ft.  9'  8'^  ^V  and  Oft.  6'  4"  ivr 


V     duodecimal  ad 
the  dition. 


SUBTRACTION. 

23'J.  Example  1. — What   is ,  the   difference  between 
'.)K.  3'  5^'  Q"'  and  7ft.  3'  C'  1'''. 

OPKRATION.  I  EXl'LAN'A  HON'. 

As  the  G  of  the  minuend 
is  le.ss  than  the  7  to  be  sub- 
tracted from  it,  we  add  13, 
and  say  7  I'rom  !  >^==  1 1. 
That  put  down  and  carry- 
ing 1,  for  the  borrowed 
number,  to  the  6,  we  say  7 


From  -0 
Take    7 


12 
3^ 
3 


12 
5' 
G 


12 

7 


6 


Rem.    lft.4r  10^ 
Proof,  9       3      5 

from  17,  which  is  the  5  added  to  12,  and  put  down  the 
remainder,  10,  and  so  on. 

2.  What  is  the  difference  between  40ft.  G''  6"  and  29ft. 

3.  What  is  the  difference  between  12ft  7'  9^'  CV'  and 
4ft.  9'  V  0'''?  Ans.  .7ft.  10'  T'  9^'^ 

4.  What  is  the  difference  between  'ISft.  8'  T'  G""'.  and 
12ft  7'  8"  b'''? 

5.  What  is  the  difference  between  19ft.  9'  S''  1''',  and 
14ft.  5' 9''  8'^'? 

6.  What  is  the  difference  between  30ft.  8'  d''  10''^  and 
?9ft..  9'  8''  11''^? 


120  *  ■     DUODECIMALS. 


MULTIPLICATION. 

how  muitipii-  233.  The  multipiioatioa  of  decimals  consists  in  niulti- 
Jbrined!^  ^^^'  V^P'^S  ^^^^^  ^^^'^  ^^'  the  multiplieaud  by  each  term  of  the 
multiplier,  couimeucing  with  the  highest  unit  of  the  mul- 
tiplier and  the  lowest  of  the  multiplicand,  and  making  the 
indices  of  each  product  equal  to  the  sura  of  the  indices  of 
the  factors. 

Example  l.—A  board  is '5ft.  b'  4^^  in  length  cVid  3ft. 
5^  4^'  in  breadth  :  nvhat  are  the  contents  ? 

Ans.  18ft.  9'  5'''  \f''\ 

Kxplanation  of      -        ^'t     ^'t.  !  EXPLANATION. 

work  of  rnuiti-     o       D      4  i "     In  this  work,  as  the  4  'X 

plication.  ,       3         5/      4//  I    g£^__22^'---l'  (12-r-r2— 1), 

we  set  down  0  under  the 
seconds,  and  add  the  1'  to 
the  next  product  of  5'X3ft. 
=16',  which,  reduced  by 
dividing  by  12'F=lft.  4',  we 
then  write  down  the  4'  and 
carry  the  1  to  the  next  prod uct=16fb. 

In  the  same  way,  we  multiply  the  multiplicand  by  the 
next  figure,  5',  and  this  line  being  written  down,  we  mul- 
tiply by  the  4  in  like  manner.  The  products  are  then 
added  for  the  answer. 

2.  How  many  square  feet  in  aboard  17ft.  6in.  long  and 
1ft.  7  inches  wide  ^  Ans,  27ft.  8'  6". 

o.  What  quantity  of  boards  will  it  take  for  a  floor  14ft. 
8'  3"  long  and  13ft.  6'  9"  wide? 

yi».s-.  199ft.  2*  4"  8'"  3'";. 
4.  How  many  feet  in  a  plank  l2ft.  4'  Ions',  2ft.  3'  wide, 
,ind4'thickr'  •     ^Ans.  111ft. 


OrEHATlON. 

12 

12 

0 

5^ 

4// 

3 

6' 

4// 

- 

10 

4' 

Q'' 

2 

.'V 

?/'' 

g/// 

V 

iV 

9/// 

4//// 

18ft. 

9' 

0" 

5'"; 

4"" 

12 
2 

QPKnATIOX. 

4'           length. 
3'           width. 

24 
3  - 

8' 
1' 

0" 

27 

9' 

4' 

0" 

thickness. 

11 1ft.   Q'      0" 


DUODECIMALS. 


121 


\  5.  What  are  the  couteuts  of  a  block  of  marble  that  is 
ft.  9'  3"  loHg,  3ft.  2'  4"  wide,  and  2ft.  5'  7"  thick  ? 

Am.  60ft.  0'  10"  4'"  5""  1'"". 
0.' What  number  of  cubic  feet  in  a  granite  block  3ft. 
;/^i.  wide,  2ft.  Sin.  thi6k,  and  12ft.  Gin.  loui?? 

Ans.  105ft.  6'  7"  6'". 
How  many  square  j'ards  in  the  walls  of  a  room  14ft. 
lonsTj'llft.  Gin.  wide,  and  7ft.  llin.  high  ? 

Alls.  46jd.  3'  8". 
What  will  it  cost  to  plaster    a  room  20ft.  G'  long, 
wide,  and  9ft.  6'  high,  at  24  cents  per  scfuurc  yard  ? 

Ans.  $25.18f. 
How  many  square  yards  of  oil  cloth  will  it  take  to 
an  entry  that  is  IGft.  Bin.  long  and  8ft.  7in.  wide  ? 
1^.  If  a  load  of  wood  be  8ft.  long,  3ft.  9in.  wide,  and 
It.  pin.  hiiih,  how  much  does  it  contain  ? 

Ans.  Ic.  4c.  ft.  3cu.  ft. 
Ill  How  many  feet  of  boards  will  it  take  to  make  12 
uuxeA  whose  interior  dimensions  are  4ft.  5',  3ft.  G',  and 
/  2ft.  7ui.,  the  boards  being  1'  thick  ? 

12.uIow  many  solid  feet  in  a  stick  of  timber  25ft.  Gin, 
t.  7in.  broad,  and  3ft.  3in.  thick  ? 

13.  \Iow  many  cords  in  a  pile  that  is  25ft.  7'  long,  5ft. 
4in.  hiji,  and  2ft.  9'  wide  ?  . 

14,  \\hat  will  a  marble  slab  cost  that  is  7ft.  4'  long  and 
1ft.  Sin, \vide,  at  §1  per  foot?  ,A?iS.  ^9. IG 3. 


in  VISION. 

2JJ4»  E>qfLMPLE  1. — In  1G5'  how  many  feet? ' 

lG5-^12==13ft.  9in.  Ans. 

2.  In  2G0\  how  many  feet  and  inches  ? 

3.  In  43G^'"  how  many  feet  ?        Ans.  2ft.  6'  3"  11'"- 

OI'BUATIDN. 

12)43G7""     ■    ■ 


12)363.  11'" 
12)30.    3" 


4.  In  5280'"  h 

5.  In  28800 
G.  In  5G450 


2.    6' 
many  feet  ? 
tw  many  feet  ? 
\w  many  feet '/ 


122  .    ANALYSIS 


ANALYSIS. 

Analysis  de-  235.  Anali/ah,  in  arltlimetic,  is  the  separation  of  a  svai 
into  its  component  parts,  with  the  relative  bcariags  of  he 
numbers  in  that  question  to  each  other. 
It  dispenses  230.  It  (^ffers  u  .simple  and  practical  method  to  pcrf'rm 
nOes'^"'^"^*'  an  operation  without  the  formality  of  rules  j  and  it  is  •  ery 
useful  in  aiding  ^lutions  when  rules,  as  in  practical  qac8- 
tions,  occurring  daily,  are  not  remembered,  and  cannoG  be 
conveniently  referred  to. 

237.  1q  analysis  we  reason  gtcp  by  step  from  the  in- 
quiry to  the  result. 

Example  1. — If  6cwt,  of  hay  cost  $13,  what  will  Qcwt. 

cost?  ^    ^  Ans.  m. 

Expianalion  of     The  auijysis  is  that  one  hundred  weight  costs  one  sixth 

*"'"  as  much  as  six  hundred  weight.     Since  6cwt.  cos^  12,  1 

costs  i  12=2  ;  9  costs  9  times  as  much  as  Icwt.,  a'ld  that 

is  9  times  i.l2=:lh.. 

2.  If  9  men  can  dig  a  trench  in  25  days,  how  mmy  will 
it  take  to  do  the  work  in  5  days  ? 

3.  A  ship's  company  have  .provisions  to  last  J2  men  8 
months  :  how  long  would  these  last  18  men? 

4.  A  hare  has  39  rods  the  start  of  a  houni,  but  the 
«N          l^outid  runs  27  rods  while  the  hare  runs  24 :  how  many 

rods  must  the  hound  run  to  overtake  the  haie  ? 

Ans.  Bfil . 

5.  The  United  States  commander  in  Fort  Sumter  bad 
21b.  of  bread  per  day  for'  each  soldifer,  for  10  days;  but, 
by  private  dispatches,  learning  that  his  goveinmeut  would 
relieve  him  soon,  he  wi.shes  to  stave  off  surrender  15  da3's : 
to  do  that,  what  must  be  the  daily  allowf'i^^e,  say  to  80 


men 


6.  24  is  I  of  what  number  ?         ' 

7.  70  is  -i%-  of  what  number  ? 

8.  If  ^  of  a  cask  of  wine  cost  $54,  what  will  4  casks 
cost  ? 

"  9.  A  man  sold  ,a  watch  fpr  $56,  which  was  |-  of  its  coat : 
<iwhat  did  he  gain  by  the  sale  ? 

10.  A  farmer  being  asked  the  number  of  liis  sheep,  said 
that  if  he  had  as  many  more,  ^  as  many  more,  and  2^,  he 
would  have  a  hundred  :  how  many  did  he  have  ? 


ANALYSIS.  ,  128 

n,   A  pole  is  -f  In  the  mud',  ^  in  the  ,\pater,  and  G  feet 
above  the  water  :  what  is  the  length  ? 
;     li.  Three   men  hire  a  pasture  for  S()0 ;  A  puts  in  2 
horses  o  weeks,  B  6  lu)rses  2^  weeks,  C  9  horses  li  weeks  : 
what  ought  each  to  pay?     Aiis.  A  $12;  1>  S<>0  ;  C  S-4. 

ANaItSIS  OF  Tim    ABOVK. 

The  pasturage-  of  2  horses'  for  v}  wceka  is  the  ■same  as  Explanation  by 
that  of  I  horse  2  times  o  weeks,  or  (\  weeks;  that  of  (> '^"■'^y-^'s- 
horses  2^  weeks,  the  same  as  that  of  1  horse  (i  times  2} 
weeks,  or  lo  weeks  ;  and  tliat  of  9  horses  H  weeks,  the 
Bamc  iis  that  of  1  horse  !)  times  lis,  or  I'J.  weeks.  The  ad- 
dition of  the  weeks  together.  ()4-l;'i4-l-=o3  weeks ;  hence, 
A'.s  share  of  payment  is  -jVof  S'>1>  ('56^o:J=:iXU^Sl2)  ; 
"s  i^  of  $6(3— f.:5Jj  and  O's  i|  of  S()t)=:§24. 


Remark. — The  pupil  should  begin  an  analysis  from  the 
term  which  is  of  the  ftame  name  or  kind  as  the  required 
answer. 

lo.  The  inheritor  of  an  estate  spent  i  of  it  in  9  months, 

and  -f  of  the  remainder  in  12  mouths  more,  when   he  had 

only  iJt'WO  left :  what  was  the  estate  when  he  received  \t  ? 

\.   Divide  $l7i).40  among  .3-  persons,  so  that  A  shall 

twice  as  much  as  B,'  and  0  three  times  as  much  as 

what  is  the  am^junt  when  so  divided  ?   ' 

■  >    Two  men  had  the  Siinie  income.     The  first  saved  i 

of  his  each  year  ;  but  the  second,  by  spending  §20(t  a  year 

more  than  the  first,  w«s,  at  the  end  of  5  years,  $Wb  in 

debt;  what  was  the  income?  Aiif^.  Slo-it. 

K).  If  it  take  44  yards  of  cai^peting,  liyd.  wide,  to 
dover  a  floor,  how  manv  yards  of  the  kind,  i  wide,  will  it 
take  ?  '  Ans.  (i.f . 

17.  If  an  acre  of  land  cost  ^  off  of -f-  of  350,  what  will 
8 J  acres  co.st?  Ann    §10. 

18.  If  :5U  gallons  of  ale  are  worth  89f,  what,  at  that 
rate,  will  5i  cost?  .  .        Ans.  $!l.G(). 


PROPORTIONAL  ARITHMETIC. 


PART     FOURTH. 


RATIO   AND  PIIOPOKTION,   OR  SIMPLE  RULE 
OP  THREE. 


Ratio  defined. 


M  umbers  rela- 
tivuly  viewed. 


]{iiw  ratio  is  in- 
diciued. 


Tlie  terms. 


Th<»  Qnt<5ce- 

dents. 

The  cocsc- 

'luents. 

First  and  see- 

I'md  couplets. 


Tlie  work  of  a 
l^roi)ortion. 


X  ratio  of  equal- 
ity. 


t  >f  'sreater  ine- 
iinafity ;  of  le.ss 
iuo'iuality. 


238.  Ratio  is  that  r'elation  of  one  quantity  or  number 
to  anotlier  of  the  same  kind,  by  which  is  found  their' 
equality  or  inequality. 
■  239.  When  numbers  are  relatively  viewed,  the  one 
which  measures  the  other  is  considered  as  the  standard, 
arid  the  quotient  resulting  from  the  division  by  the  stand- 
ard is  the  ratio  or  relative  value. 

240.  Ratio  is  indicated  by  a  colon  (:)  in  the  first  term ; 
by  a  double  colon  (::)  in  the  second;  and  by  a  colon  in  the 
third  (:);  thus, 

'  4:8::G:12  ;  that  is,  4  is  to  8,  as  6  is  to  12. 

24L  The  two  quantities  compared  are  the  terms  of  the 
ratio  ;  the  first  terms,  or  4  and  6,  'in  the  above  expression, 
being  the  antecedents,  and  the  second,  6  and  12,  the  con- 
sequents. Here,  the  antecedents  are  the  standard.  The 
4  and  8  are  known  as  the  first  couplet,  and  'G  and  12  the 
second. 

242.  When  two  couplets  have,  as  in  the  example,  the 
same  ratio,  their  terms  are  proportional ;  hence, 

243.  A.  proportion  compares  the  terms  of  ts5;o  equal 
ratios. 

244.  When  the  antecedent  equals  tlrtj  consequent,  the 
ratio  being  one,  it  is  called  a  ratio  of  equality;  thus,  4:4 
=  1. ;  when  it  is  greater,  or  more  than  One,  it  is  known  as 
a  ratio  of  greater  inequality;  thus,  6:2=3;  when  it  is  less 
than  one,  a  ratio  of  less  inequality ;  thus,  2:lC=i. 


RATIO  AND   PROl'ORTION.  125 

245.   When  there  is  but  one  antecedent  and  one  conse-  a  simple  ratio, 
quent,  the  ratio  is  said  to  be  simple;  thus,  12:4=3. 

2146.  When  tlie  corresponding  terms  of  two  or  more  a  compomui 
simple  ratios  arc  multiplied  together,  the  ratio  arising  from '^''^'' 
it  is  called  compound  ;  thus, 
4:  2=  2 
4:2^2  6:  3=  2 

G:2=3  12:  4=  3 


24:4=6  and  288:24=12  are  compound  ratios ;  and,  .as 
the  illustration  shows,  are  equal  to  the  product  of  the  sim- 
ple ratios  of  whicli  tliey  are  composed. 

24'/.  The  1st  and  4th  terms  of  a  proportion  are  called  t^^p  extremes; 
•the  extremes;  the  2d  and  3d  the  means;  and  the  product  ^^  '"eans. 
of  the  extremes  is  equivalent  to  ihat  of  the  means  ;  thus, 
in  the  proportion, 

3:9::12:36;  3X36=-- 108;  and  0x12=108.  .\ 

248.  The  4th  term  of  a  proportion  is  found  by  multi-To  find  (he 
plying  the  second  and  third  terms  together,  and  dividiug  ""  ''    '^'^"'' 
by  the  first;  thus,  3:9::12:  ?;    9x12=108-4-3=36,   the 
term  sought. 

•    Example  1. — What  is  the  4th  term  in  the  proportioii, 
3:  9::  4  ? 
5:15::  3  ? 
6:  8::12  ? 
8:  4:-:  4? 

Remark. — Any  term  of  a  proportion   can   be  found  To  find  other 
when  the  three  others  are  given  ;  for  the  product  of  the  '^'""'*- 
extremes,  divided  by  either  mean,  gives  the  other;  and 
the  product  of  the  means,  divided  by  either  extreme,  gives 
the  other. 

In  the  solution  of  problems,  two  of  the  three  given  num-T^.f,r,ft,,<,Qu„,. 
bers  must  be  of  the  same  kind ;  the  third  like  the  one  re-  bers  to  bo  of 

,        ,  '  correspondent 

quired ;  thus,  n.sturc. 

2.  If  4  men  build  8  rods  of  wall  in  a  day,  how  many 
will  B  |)uild  ?  Ans.  12. 

4  m^n  :  6  men  ::  8  rods  :  12  rods,  the  number  sought. 

3.  If  a  staff  6ft.  long  cast  a  shadow  12  feet,  what  shadnvv 
will  be  cast  at  the  same  time  by  a  steeple  60  feet  high  ? 

4.  If  a  staff  6  feet  long  cast  a  shadow  9  feet,  what  is  the 
hei^-ht  of  a  tower  w;hose  shadow,  at  the  same  hour,  extenus 
198  feet  ?  and  what  is  the  ratio  ? 

Ans.  132ft. ;  Ilatio,  22. 


126  RATIO   AND   PROPORTION. 

5.  If  board  for  52  w'eeks  amounts  to  $182,  what  is  it 
for  39  weeks?  Ans.  ^iHQ/oO. 

G.  If  I  pay  for  48  yards  of  cloth  S87.7->,  what,  at  the 
same  rate,  will  144  cost?  Aiis.  S-Ol.T;^. 

7.  If  S'O  soldiers  require  11,250  rations  of  bread  for 
a  month,  how  many  will  be  necessary  for  a  garrison  of 
600?  Ans.  18,0G0. 

8.  If  12  men  can  build  a  house  in  20  days,  how  many 
can  do  the  work  in  5  days  ?  and  what  is  the  ratio  ? 

Ans.  48  men  ;  Ratio,  4. 

9.  If  80  gallons,  in  an  hour,  run  into  a  reservoir  tbat 
will  contain  1400,  and  30  run  out,  in  what  time  will  it  be 
filled?  '  -^"s.  28  hours. 

Soiuiions  by         Eemark. — A  uscful  Avay  ,to  solvc  qucstions  like  these 
analysis  prefer- is  by  analysis.     As  in  busiue^ss  operations  such  method  is 
^'  usuully  adopted,  we  would  adKse  it  as  practically  the  bet- 

ter plan. 

10.  3  bricklayers  build  6  rods  of  a  inundation  in  a  day.: 
how  many  rods  would  5  build  ? 

If  8  men  build  6,  1  will  build  J  of  6=2  j  if  1  build  2, 
5  will  build  5  times  2=10. 

11.  If  25  men  perform  a  certain  work  in  35  days,- how 
long  will  it  take  9  to  do  the  same  ? 

12.  If  271b.  of  butter  will  buy  451b.  of  sugar,  bow 
much  can  be  had  for  SOibs.  of  butter? 

13.  If  an  engineer's  salary  for  throe  years  amount  to 
$3600,  what  will  it  be  in  9  years  ? 

14.  Iff  of  a  yard  ol  cloth  cost  f  of  a  dollar,  what  will 
21  yards  cost?  ^«s.  S4.8ij. 

Here  we  say,  if  f  cost  -f-,  ^  costs  A  of  i=-(Vx5  for  the 
whole  =ffX2J,  that  is  -I  =-Yb^)  which,  by  the  annexing 
of  two  ciphers  ibr  cents,  =4.bG. 

Note. — Let  it  be  remembered  that  a  mixed  number,  as  in 
this  sum,  is  to  be  reduced  to  a  fraction. 

15.  What  is  the  price  of  Gfyd.  of  cloth,  if  '}yd.  cost 

16.  What  is  the 'cost  of  3Sdoz.  of  wine,  if  jdoz.  coat 

I7H 

17.  What  ifi  the  value  of  SSJ  barrels  of  ale,  if  4i  bar- 
rels cost  $15? 

18.  A  merchant  owning  f  of  a  ship,  sella  |  of  his  s-haro 
for  S  15,000  :  what  is  the  value  of  the  ship  ?      .. 


COMPOUND    PROrOKTION.  127 

19    llow  many  yards  of  silk,  f  yd.' wide,  will  it  l^kc  to 
line  ITiyd.  of  camlet  iyd.  wide  ?  'f^ 

20.  II',  when  flour  is  worth  ^tl  per  barrel,  a  5  cent'  Idaf 
weigli  4oz.,  what  ought  it  ^  weigh  when  flour  is  Worth    " 
$8  per  barrel  ?  ,  "  i' 

21.  If  the  earth  revolve  366  times  in  365  days,  in  what 
time  does  it  revolve  ojx^e?  Aus.  23h.  SG-jVm. 

22.  If  ,62a.  3r.  2pi:  of  land  cost  £615  93.  3d.,  what  will 
1 5a  2r.  3rd.  cost  7  -      ' 

23.  If-,V  ofaship  cosfctl350,  what  isf  of  her  worth? 

24.  IJorroved  $25l)  'for  6  months  ;  for  how  long  must 

$35J  be  le«t  to  repay  the  favor  ?  •       • 

25.  Ii"l2iyd.  of  silk,  Jyd.  wide,  will  make  a  dress,  how 
many  yards  of  cambric,  that  is  1|  wide,  will  Hue  the  same  ? 

Aus.  6-(*r- 


COMPOUND   PROPORTIOX,  OR  DOUBLE   RULE 
OF  THREE. 

249.  Compny.nd  Proportion  is   an  equality  which  em-  compounu  pr»- 

braces  in  its  consideration  simple  and  compound  ratios ;  portion defmod 

thus,  ^ 

3-12) 
1^'  .,  \  ::18:9  is  a  compound  proportion,  and 

4S:24::18:9  is  the  same  in  a  simple  form. 

250.  Que.stidns  can  be  reduced  to  a  simple  form,  and 
aome  of  complicated  expression  are  easily  solved,  when  the 
terms  have  been  carefully  arranged. 

ExA.Mi'LE  ].— If  3  men  in  4  hours  can  thresh  15 
bushels  of  ric6,  in  how  ;uany  hours  can  2  thresh  .') '( 

Aiiti.  2. 
^1/  Simple  Proj)ortion. 


OPERATION. 


KXrLAXATION 


It  will    be  Peen   that  inj;,p,^„^^^„  ^ 
the    first     proportion     the  qucetion 


2:3::4:6,  and 
15:rK:():2,  An?. 
amount  of  labor  i.^  not  made  the  subject  of  incjuiry,  but 
the  time  that  2  men  will  take  to  do  the  work  of  '6  men. 
Finding  that  to  be  6  hours,  the  second  iiujuiry  is,' in  what 
time  5  bu.ihels  can  be  threshed,  if  l.>  bu.^!»cls  are  .in  G 
hours,  and  the  time  i.s  found  to  be  2. 


128 


COMPOUND   PROPORTION. 


Why  called 
compound. 


How  to  solve 
questions. 


2.  If  poo  gain  $6  in  12  mobths,  what  will  be  the  gain 
of  $3400  in  18  months  ?  Ans.  $306. 

OPERATION.  I  EXPLANATION. 

100:3400  ::  6:204  and  Here  are  stated  three  of 

,12  :  18::204:306  T  the  given  things  in  a' sim- 

ple proportion  ;  that  $100  are  to  |-3400  as  $6,  the  interest, 
are  to  the  answer:  or,  if  $100  gain  $6,  what  will  $8400 
gain  ?  100:8400::6:204.  This  being  done,  there  remain 
the  terms  12  months  and  18  months  to  be  disposed  of, 
and  forming  a  second  proporti'on  under  tue  first,  we  say, 
12  months  are  to  18  months::204  (the  interest  on  3400 
for  6m.)  :306  the  answer.  . 

The  reason  why  questions  conducted  thtis  aio  called 
compound  proportion  is  evident,  for  the  answer  sought  is 
not  only  in  proportion  to  the  principal,  but  also  in  propor- 
tion to  the  time;  and,  therefore,  in  the  compound  pro- 
portion of  the  interest  multiplied  by  the  time.  » 

To  solve  questions  in  compound  proportion,  make  for 
the  3d  term  that  which  is  of  the  same  kind  or  denomina- 
tiqn  with  the  answer.  Then,  take  any  two  of  the  remain- 
ing terms  that  are  alike,  and  arrange  them  as  in  simple 
proportion.  In  a  similar  way,  arrange  any  other  two  terms 
of  the  same  kind,  and  multiply  the  continued  product  of 
the  2d  terms  by  the  3d  term,  and  divide  this  result  by  the' 
continued  product  of  the  1st  terms ;  the  quotient  will  be 
the  term  required. 

3.  Ten  men  can  build  25  rods  of  fence  in  6  days :  how 
many  men  will  it  take  to  build  30  rods  in  3  days  ? 

Ans.  24. 


OPERATION. 

25:30::10:12  and 
3:  6::12:24  Aiis. 


EXPLANATION. 

With    the   statement    of 
the  first  three  terms,  which. 


are  in  direct  proportion,  we  have  the  two  terms,  6  and  3, 
placed  in  what  is  called  an  inverse  proportion  to  the  an- 
swer, and  in  this  way,  because  the  more  men  are  engaged, 
the  less  time  is  needed.  These  terms  are  therefore  to  be 
put  inversely  from  the  order  necessaryjif  the  proportion 
had  been  direct.     Instead  of  6  to  3,  say  3  to  6. 


wiien  the  con-     Note. — This  same  rule  applies  when  there  are  three  or 
three  or 'more,  niore  conditions  to  the  question. 


4.  If  a  family  of  6  persons  spend  $300  in  8  months, 
what  amount  is  necessary  to  a  family  of  15  persons  for  20 
months?  '  Ans.  $1875. 


COMPOUND    PROPORTION. 

5.  If  a  family  of  6  persons  spend  §600  in  8  months, 
how  many  dollars  will  be  required  for  a  family  of  10  per- 
sons in  14  months  ?  Ajis.  §1750. 

6.  A  trader,  with  a  capital  of  $300,  gained  $75  in  3 
months  :  how  much  would  he  gain  at  the  same  rate  with  a 
capital  of  $1000,  in  1  year  (12  months)? 

7.  If  10  acres  feed  15  head  of  cattle  for  20  days,  how 
many  acres  would  feed  400  head  for  90  days  ? 

8.  If  50  men  can  dig  a  ditch  100  yards  long  and  4  wide 
in  30  days,  how  many  can  dig  one  400  yards  long  and  5 
wide  in  5  days  ? 

ft.  A  ship's  company  of  16  use  Ihhd.  of  water  in  1 
month  (30  days),  how  long,  if  the  number  be  increased  to 
24,  will  40hhd.  last? 

10.  If  100  men  can  build  a  wall  300  feet  long,  2  feet 
deep,  and  6  feet  high,  in  10  days,  in  how  many  days  can 
50  men  build  50  feet  of  wall,  3  feet  deep  and  4  feet  high? 


129 


stated  thus,  300 
2 

6 
50 


Remark. — This  somewhat  complicated  question  may  be 
"    50::10: 
3 
4 
100.     The  terms  are  in  direct  proportion 
to  the  answer,  because  more  requires  more ;  that  is,  the 
longer  the  wall,  the  longer  the  time  for  its  completion  j 
the  deeper  the  wall,  the  longer  time,  and  the  higher,  the 
longer  time.     The  last  proportion  is  inverse,  because  more 
men  need  .less  time,  and  it  is  the  only  one  to  be  stated 
inversely. 

11/  If  a  twopenny  loaf  weighs  8oz.  when  wheat  is  68. 
'9d.  per  bushel,  how  much  bread  may  be  bought  for  3s.  4d. 
when  wheat  is  selling  13s.  6d.  per  bushel  ?      Ans.  51b. 

12.  If  40  men  can  dig  a  trench  40  yards  long,  8  deep, 
and  8  yards  wide,  in  8  days,  how  many  men  would  be  em- 
ployed to  finish  a  trench  100  yards  long,  12  wide,  and  3 
deep,  in  24  hours  (1  day) ? 

13.  If  25  persons  consume  300  bushels  of  corn  in  1 
year,  how  many  bushels  will  139  consume  in  8  mouths,  at 
the  same  rate  ?  Ans.  1112. 

14.  If  the  cost  per  railroad  of  12cwt.  3qr.,  for  400 
miles,  is  $57.12,  what  will  be  the  cost  of  10  tons  for  75 
miles?-  ,  Ans.  $168. 

10 


COMMERCIAL  ARITHMETIC. 


PART   FIFTH. 


INTEREST. 


Intereat  de- 
fined. 


Names  con- 
nected with  it, 


251.  Interest  is  a  premium  for  the  use  of  borrowed 
money. 

252.  The  money  on  which  interest  accumulates  is  called 
the  Principal;  and  the  principal,  with  its  interest,  is 
called  the  Amount. 

Premium  dif-       253.  The  premium  is  a  legalized  value,  but  is  not  uni- 

lot^tates!^'^^'^  ^'^^^  ^^  ^^"^  Confederacy.  In  South  Carolina,  the  rate  or 
annual  per  centage  is  7  per  cent.,  that  is,  7  cents  on  the 
100  cents,  or  7  dollars  on  the  100  dollars,  for  a  year.  In 
Georgia,  it  is  8  per  cent. ;  in  Texas,  12  per  cent. ;  in  some 
of  the  States  of  the  old  Union,  6  per  cent. ;  in  France  aad 
England,  5  per  cent. 

254.  The  term  per  cent,  (per  centum)  is  Latin,  and 

"^^^  ^^^"8  P^''  signifies  for  the  hundred  j  and  per  an.  (per  annum)  for 

annuia.  the  year. 


Note. — A  sum  at  simple  interest  becomes  double  in  16 
years,  8  months. 


How  long  it         *59«  Example  1. — The  interest  on  SI,  for  a  year,  is  7 
takes  to  double  gg^^g  [^  South  Carolina,  what  is  the  amount  of  interest 
due  on  $100  for  the  same  period  r  Ans.  §7. 


an  asBount. 


OPERATION. 

Multiply         100 
By  7 

700 


Eemark.— The  $100x7 
=700;  and  as  the  multipli- 
cation of  dollars  by  cents 
(Art.  121,  Remark)  gives 
cents,  the  answer  is  700  cts. 


INTEREST.  181 

In  multiplying  dollars  by  dollars,  let  the  pupil  remem-  When  doiiflrs 
ber  that  the  result  will  be  dollars;  but  when  dollars  arc'^,^,''J^Jf|,'r'£''B^'i 
multiplied  by  cents,  as  in  the  above  example,  the  answer  doUnrs  by 
is  in  cents.     Two  right  hand  figures  pointed  off  for  cents,  '^ 
show  the  answer  to  be  87. 

2.  What  is  the  inte-rest  of  $50,  for  1  year,  at  7  per 
oent.?  Am.  §3.50. 

OPERATION. 
50 

7 

350  cents. 

3.  "What  is  the  interest  of  §90,  for  1  yeaj:,  at  7  per 
cent/!*  Ans.  S6.30. 

4.  What  is  the  interest  of  §99,  for  1  year,  at  7  per 
cent.  ? 

5.  What  is  the  interest  of  6250,  for  lyr.  6m.,  at  7  per 

cent.?  ^ 

Note. — The  interest  for  Gm.  (iyr.)  is  to  be  added  to 
that  of  the  year. 

6.  What  is  the  interest  of  S300,  for  lyr.  4m.,  at  7  per 
cent.?     (4m.=  J  of  a  year.)  A7is.'$2^. 

When  the  number  of  months  is  an  equal  part  of  12,  as  How  in  ?o,w 
in  the  last  example  (12-f-o=4).  it  facilitates  calculation  f.'**'^^  c^icu;:,- 
to  take  an  equal  part  of  the  year's  interest,  according  toted. 
the  question,  and  to  add  it  to  the  amount  for  the  year. 

Note. — The  same  applies  to  parts  of  a  month  and  to 
days. 

7.  What  is  the  interest  of  §700,  for  2  years  and   2 
months,  at  7  per  cent.  ?     (The  2m.=  h  12.) 

8.  What  is  the  interest  of  S900,  for  1  year,  3  mouths 
and  15  days,  at  7  per  cent.  ?     (The  15d.=  J  mouth.) 

A71S.  e80.37J. 

9.  What  is  the  interest  of  81000,  for  2  years,  1  month 
and  20  days,*at  7  per  cent.  ?     (The  20  days  is  fm.) 

10.  What  is  the  interest  of  375,  for  lyr.  5jn.  and  18d., 
at  7  percent.?     (5m.=T^ofl2;  ]8d.=-J^  of  30.) 

11.  What  is  the  interest  of  ^15G,  for  2  years,  8m.  and 
lOd.,  at  7  per  cent.  ?     (8m.=5  12 ;  lOd.  ^  30,  or  jm.) 

12.  What  is  the* interest  of  817.49,  for  lyr.,  7m.  and 
Sd.,  at  7  per  cent.  ?  Ans.  U.QSb. 


INTEREST. 

When  there  are     Note. — Here,  dollars  multiplied  by  cents  give  cents,^uf 

cents  in  a  quee- ggjj|.g  jj^jtipiieij  j^y  cents  givc  or  reduce  to  mills  (Art. 

121,  Renjark).    Thus,  the  answer  is  $1.  68c.  5m.=  J  cent.) 

13.  What  is  the  interest  of  $459,  for  5yr.,  at  8  per 
cent.?  '  Ans.  183.60. 

14.  What  is  the  interest  of  $600.50,  for  3yr.,  at  8  per 
cent.?  ^ns.  $144.13,  4  mills. 

15.  What  is  the  interest  of  $62,  for  3yr.,.at  8  per  cent.  ? 

16.  What  is  the  interest  on  a  Confederate  State  bond, 
for  $1000,  for  2yr.  6m.,  at  8  per  cent.  ? 

17.  What  is  the  interest  of  $156,for  ]yr.  9m.  and  15d.y 
at  12  per  cent.  ?     (9m.=  f  12m.) 

18.  What  is  the  interest  of  $256,  for  6m.  20*,  at  8 
per  cent.  ?        • 

19.  What  is  the  intei:est  of  $444,  for  lyr.  10m.,  at  6- 
percent.?     (10m.=il  of  12.) 

•  20.  What  is  the  interest  of  $85.30,  for  1  yr.  16d.,  at  6 

per  cent.  ? 

21.  What  is  the  interest  of  $550,  for  90  days,  at  7  per 
cent.  ?     (90d.=3m.)  ' 

22.  What  is  the  interest  of  $150,  for  60  days,  at  t  per 
cent.?     (60d.=2m.) 

23.  What  is  the  interest  of  $125,  for  30  days,  at  7  per 
cent.?     (30d.=  lm.) 

24.  What  is  the  interest  of  $38.55,  for  2yr.,  at  7  per 
cent.  ? 

What  to  kc^'p  Remark. — Let  the  pupil  remember  that  when  cents  are 
'"^oi^^nc  off  i^  *^6  question,  4  figures,  counting  left  to  right,  are  to  be 
Dumbefs.  pointed  off;  and  that  the  two,  on  the  right  of  the  point, 

are  cents,  and  the  others  tenths ;  thus,  as  in  the  14th  ex- 
ample, the  numbers  in  the  result  are  1441344,  which, 
pointed  off  as  directed,  read  $144.13  cents,  \^,  or  4 
;(uills,  etc. 

25.  What  is  the  interest  of,  $95.50,  for  lyr.  6m.,  at  7 
per  cent.  ? 

26.  What  is  the  interest  of  $350,  for  3yr.  8m.  18d., 
at  6  per  cent;,  ?  Ans.  $79,388. 

JIemArk. — ^In  mercantile  transactions  it  is  customary  to 
take  for  the  answer  only  two  figures,  cents,  at  the  right 
hapd  of  the  separating  point.     This  answer  would  usually 

b?i:ead  $79.38. 


INTEREST. 

27.  What  is  the  interest  of  $326,  for  3yi-.  2m.,  at  7  per 
«ent.  ? 

28.  What  is  the  interest  on  $56.52,  from  March  19, 
1859,  to  January  25,  1862,  at  7  per  cent.  ? 

29.  What  is  'the  interest  on  §598,  from  July  15,  1860, 
to  Oct.  20,  1864,  at  7  per  cent.  ?     ■ 

30.  What  is  the  interest  on  $135,  from  Dec.  10,  1861, 
to  May  17,  1868,  at  7  per  cent.  ? 

31.  What  is  the  interest  on  $65.^,  from  Jan.  1,  1855, 
to  Feb.  1,  1863.  at  7  per  cent.  ? 

32.  What  is  the  interest  on  8100,  from  June  10,  1860, 
to  June  10,  1865,  at  7  per  cent.  ? 

33.  What  i.s  the  interest  on  £27  15s.  9d.,  for  1  year,  at 
5  per  cent.  ?  Ans.  £1  7s.  9d.  Iqr, 

OFERATIOX. 

£27  15s.  9d.=27.7875 

5 

1.389375=£1.389375 
20 


133 


d.  9.4500 
4 


EXPLAXATION. 

As  the  principal  is  in  pounds,  shillings,, 
pence,  we  reduce  the  lower  denominations 
to  the  decimal  of  a  pound  (Art.*  214,  14 
Ex.),  and  multiply  that,  with   the  pounds 
prefixed,  by  the  per  centage.^   In  this  ex- 
ample, the  left  hand  figure  is  the  interest 
of  the  pounds  denomination.     The  decimal 
interest  is  then  reduced  back  to  shillings,     qr.    1.80 
pence  and  farthings.     It  will  be  noticed  that  only  3  deci- 
mal places  in  the  multiplicand  ^re  used. 

34.  What  is  the  interest  of  .£16  9s.  5d.  2qr.,  for  lyr. 
6in.  15d.,  at  5  per  cent.  ? 

35.  What  is  the  interest  of  £75  123.  9d.  3qr.,  for  2yr. 
9m.  20d.,  at  5  per  cent.  ? 

36.  What  is  the  interest  of  £100  3s.  3d.,  from  March 
9,  1862,  to  June  24,  1864,  at  5  per  cent.  ? 

,    37.  What  is  the  interest  of  £90  6s.  3d.,  from  Dec.  1, 
1862,  to  Oct.  11,  1863,  at  5  per  cent.  ? 

38.  What  amount  of  principal  and  interest,  at  7  per 
ceftt.,  is  due  Jan.  1,  1861,  on  a  note  of  S400,  dated  Jam 
1,  1859;  there  having  been  paid,  July  1,  1859,  8100; 
Jan.  1,  1860,  $150  ;  and  July  1,  1860,  $50  ? 

Remark. — In  like  examples  cast  the  interest  from  the  wh^H  naf- 
dt^e  of  the  note  to  the  specified  time,  and  add  to  the  prin-  b"^u'm3.iJo«  & 
flipal ;  then,  on  the  several  payments,  from  their  dates  to  *»ote. 

9 


lo4  INTEREST. 

the  specified  time,  and  add  to  their  collected  amount  this 
credit  interest ;  subtract  the  same  from  sum  of  principal 
and  interest,  for  answer. 

39.  What  was  due  on  a  note  of  $448.50,  at  7  per  cent. 
interest,  dated  June  15j  1854,  when  finally  settled  July  3, 
1856,  which  had  these  endorsements  :  Dec.  6,  1854,  $75  : 
April  19,  1855,  $125;    Dec.  15,  1855,  $10;    Jan.  25, 

•       1856,  $100?  .  .'  "    •  Ans.  $183.60. 

40.  What  was  due,  March  26,  1855,  on  a  note  for 
$1000,  at  6  per  cent,  interest,  on  which  had  been  paid, 
Sept.  6,  1850,  $50  ;  July  14,  1851,  $150 ;  Aug.  9,  1852', 
$25  ;  May  14,  1853,  $28  ;  Oct.  15, 1853,  $125  ;  Nov.  11, 
1853,  «75  ;  and  Not.  13,  1854,  $500  ?     Ans.  $282.58. 

To  find  the  jf  it  j^g  desirable  to  ascertain  the  principaL  when  the 

iinncspal  when  ,  .  i.i  i  "^  ■,      •    ,  x 

t)i9  interest,     interest,  time  and  rate  alone  are  known,  cast  the  interest  on 
^'i"\nown.'''^  one  dollar  for  the  given  time,  and  divide  the  interest  by,  it. 

41.  The  interest  of  a  certain  sum,  for  2  years,  at  7  per 
cent.,  is  $70  :  what  is  the  principal  ?  Ans.  $?00. 

OPERATION. 

Int.  on  $1=7X2(2  years)=14;  and  70.00-f-14=500. 

42.  The  interest  of  a"  certain  sum,  for  lyr.,  at  7  per 
cent.,  is  ^63  :  what  is  the  principal  ?  Ans.  $900. 

43.  The  interest  of  a  certain  sum  is  $350  a  year,  at  7 
per  cent.  :  what  is  the  principal  ? 

44.  The  interest  of  a  certain  sum,  for  lyr.,  at  7  per 
cent.,  is  $1500  :  what  is  the  principal  ? 

'I'o  find  the  rate      When  the  interest,  time  and  principal  are  known,  we 
tlineandpritic'i- ascertain  the  rate, by  casting  the  interest  on  the  principal 
7Xii  are  known,  f^y  ^\^q  given  time,  at  1  per  cent.^  and  then  dividing  the 
known  interest  by  it.     The  quotient  is  the  rate. 

45.  The  interest  of  $500,  for  lyr.,  is  835  :  what  is  the 
rate?  Ans.  7  per, cent. 

OPERATION.  ]       Here,  to  accomplish  the  division 

500)35.00(7  of  35  by  500,  we  annex  (Art.  129) 

35.00  1  two  ciphers  to  the  S35. 

46.  The  interest  of  $1200,  for  2  years,' is  ^144  :  what 
is  the  rate  ?  Ans.  6  per  cent. 

,  47.  The  interest  of  81500,  for  1  year,  is.  $120 :  whut 
is  the  rate  ?  •  *       • 

48.  The  interest  of  82000,  for  lyr.,  is. $100:  what  is 

the  rate  ? 

To  find  tiie  When  interest,  principal  and  rate  are  known,  we  ascer- 

jldnoipai!  hiter^-  tain  the  time  by  casting  the  interest  on  the  given  pri^- 

.-st,  and  rate  are  cipal,  at  the  kuown  rate,  for  1  year,  and  dividing  the 


*  COMPOUND    INTEREST.  135 

interest  by  it.     The  quotient  will  be  the  time  in  years  and 
decimals  of  a  year. 

49    What  is  the  time  when  the  principal  is  ^900,  the 
interest  is  $63,  and  the  rate  is  7  per  cent.  ? 

Arts.  1  year. 

OPERATION. 

Int.  900=63,  and  63-=-63=l. 

7 

63.00 

50.  What  is  the  time  when  the  principal  is  $750,  the 
interest  S90,  and  the  rate  6  per  cent.  ?        Ahs.  2  years. 

51.  For  what  time  must  $200  be  at  interest,  at  6  per 
^ent.,  to  gain  $36  ? 


COMPOUND  INTEREST. 
256.  Compound  Interest  is  interest  on  principal  and  compoun.i  in- 

...  .f.    J  J  1  .  .'^    .       T^       rr,,      terest  defined. 

interest  iinited  and  making  a  new  principal.  The 
latter  being,  as  an  unliquidated  debt,  added  to  the 
former. 

.  257.  No  law  recognizes  such  computation,  but  it  is  ite  computa- 
equitable  and  becomes  legal  when  payment  of  due*'°°  equitable, 
interest  is  not  made  on  demand.     In  such  case  the 
unsettled  interest  becomes  a  principal,  and  interest, 
as  on  any  debt  similarly  placed,  can  be  charged. 

-.r  t  ,     .  1       1  1       I"  whai  time  a 

Jyote. — An  amount  at  compound  interest  doubles  Bum  at  com- 
itself  in  11  years,  8  months  and  22  days.  fs'douwJd"" 

25S.  The  method  of  computing  it  consists  in  find- Method  of  com 
ing,  as  in  simple  interest,  what  is  due  on  a  specified  ^"  *  '^"' 
amount,  added  to  that  amount,  and  then  annually 
increased  by  the  interest  of  the  preceding  year. 


loS  COMPOUND    INTEREST. 

Example  1. — What  will  be  the  compound  interest 
of  $500,  for  2  years,  at  7  per  cent.  ?        Aiis.  $72.45 

OPERATION. 

$500.00  principal  1st  yr. 
35.00  int.  for  Ist  yr.,  500x7=35. 

535.00  principal  2d  year. 
37.45  int.  for  2d  yr.,  535x7=37.45. 

572.45  amount  in  2yr. 
500.00  deduct  first  principal. 


$72.45  amount  of  interest. 

Hote. — It  will  bo  seen  that  the  whole  amount  of 
compound  interest  is  found  by  taking  the  first  prin- 
cipal from  the  last  sum  of  ijrineipal  and  interest. 

2.  What  is  the  compound  interest  on  $1000,  for  B 
years,  at  7  per  cent.  ?  Ans.  $225,043. 

3.  What  will  be  the  amount  of  $5000,  at  compound 
interest,  7  per  ceift.,  for  4yr.  10m.?    Ans.  $6911.26. 

4.  What  is  the  compound  interest  of  $250,  for  2yr., 
at  8  per  cent.  ?  Ans.  $41.60 

5.  What  is  the  compound  interest  of  $939.64,  for 
3yr.,  at  7  per  cent.?  Ans.  $211.45. 

6.  What  is  the  compound  interest  of  £50  98.  6d., 
for  2yr.,  at  5  per  cent.  ? 

Note. — Keduce  shillings  and  pence  to  decimals  of  a 
pound  (Art.  214,  14  Ex). 


COMPOUND    INTEREST. 


137 


An  expeditious  way  to  calculate  compound  interest 
is  afforded  by  this 

TABLE, 

fihowing  the  amount  of  $1,  £1,  etc.,  interest  compounded  annually 
at  4,  5,  C,  7  ajid  Sper  cent.,  from  1  to  20  years. 


4  per  cent.   |  5  per  cent,    i  6  per  cent. 


1.040000 

1.081600 

1.1248G4 

1.169859— 

1.216653— 

1.26i)319+ 

1.315932— 

1.368569+ 

1.423312— 

L480244+ 

1.539454+ 

1.601032+ 

1.665074— 

1.731676+ 

1.800944— 

1.872981+ 

1.947900+ 

2.025817— 

2.106849+ 

2.19112.3+ 


1.050000 

1.102500 

1.157625 

1.215506+ 

1.276282— 

1.340096— 

1.407100+ 

1.477455+ 

1.551328+ 

1.628895— 

1.710339+ 

1.795856+ 

1.885649+ 

1.979932— 

2.078928+ 

2.18287,5— 

2.292018+12, 

2.406619+12 

2.o26950+i3 

2.653298—1.3 


060000 

123600 

191016 

,2T]2477— 

,338226— 

,418519+ 

,503630+ 

,593848+ 

,689479— 

,790848— 

,898299— 

,012196+ 

,132928+ 

,260904— 

,396558+ 

540352— 

,692773— 

,854339+ 

,025600— 

,207135+ 


7  per  cent.   8  per  cent.   > 


070000 

144900 

225043 

310796+ 

402552—; 

500730+j 

605781+i 

718186+1 

838459+1 

967151+ ! 

104852—' 

252192— i 

409845+ 

578534+ 

759032—' 

952164—' 

158815+' 

379932+ j 

616528— i 

869684+ ! 


080000 

166400 

259712 

360489— 

469328+ 

586874+ 

713824+ 

850930+ 

099005— 

158925— 

331639— 

518170+ 

719624— 

937194— 

172169+ 
425943— 
700018+ 
996019+ 
315701+ 
660957+ 


Table  for  find- 
-  ing  oompound 
1  interest. 

2 

3 

4 


7.  AVhat  is  the  compound  interest  of  $17.25,  for 
'Jyr.  and  7m.,  at  5  per  cent.  ? 

Note. — From  the  table  take  the  amount  of  $1,  for 
2  years,  at  5  per  cent.,  and  compute  the  interest  on 
it,  for  7  months,  us  in  simple  interest.  Add  this  to 
the  amount  for  2  years,  and  we  have  the  interest  of 
$1  for  2yr.  7m.  Multiply  this  by  $17.25  for  "answer. 
To  find  the  interest,  subtract  the  principal  from 
amount. 

8.  "What  is  the  compound  interest  of  $300,  for  5yr. 
8m.  15d.,  at  6  per  cent.  ?  Ans.  $171.59. 

9.  What  is  the  compound  interest  of  $600,  at  6  per 
cent.,  per  annum,  for  20yr.  ? 

OPERATION. 

$3.207l35=int.  of  $1  for  20yr. 
600  principal. 


$1924.281000  int.  of  $600  for  20yr. 


138 


DISCOUNT. 


Here,  counting  off  4  figures  on  the  left  for  dollars^: 
to  the  unit-dollars-figures  in  the  multiplicand,  and 
the  three  of  the  multiplier,  600,  we  have  the  answer, 
$1924.28. 

10.  What  is  the  compound  interest  of  $950,  for  3yr. 
■  6m  ■  lOd.,  at  7  per  cent.  ? 


DISCOUNT. 


Discount  dO' 
fined. 


Tke  per  eent- 


To  find  the 


259.  Discount  is  an  amount  allowed  for  the  cash 
payment  of  a  bill,  or  for  the  settlement  of  a  debt, 
before  its  unexpired  term  of  credit,  or  the  sum 
charged  by  a  bank  for  money  loaned  on  a  note. 

260.  The  allowance  is  called  per  centage,  and  its 
rate  is  according  to  agreement.  When  no  rate  has 
been  named,  it  is  customary  to  deduct  or  discount 
the  interest. 

261.  Discount  is  found  by  subtracting  the  present 
amount  of  dis-^^^^^  f^^^^  ^^le  amoiint  due. 

Example  1. — What  is  the  present  value  of  $100, 
payable  at  the  expiration  of  a  year,  without  interest  ? 

Ans.  $93.45 

EXPLANATIOK. 

We  divide  the  given 
sum  by  $1,  with  its  in- 
terest for  the  time ;  jn 
this  example,  7  per  cent. 


OPERATION. 

1.07)100.00(93.45+ 
963 


370 
321 

490 

.       428 

620 
535 

85 

2.  What  is  the  present  value  of 
payable  in  lyr.  4m.  ? 


,  at  6  per  cent., 


COMMISSION.  1'^*^ 

3.  What  is  the  value  of  $500,  due  lyr.  bonce,  at  8 
per  cent.  ?  . 

4.  What  is  the  discount  to  be  made  on  a  note  for 
$437,  which  has  an  unexpired  term  of  4ni.  15d.  ? 

5.  What  is  the  present  value  of  a  note  for  $350, 
payable  in  1)0  days,  which  has  been  discounted  for 
me  at  the  Bank  of  the  State  ?  Ans.  $343.68. 

Remark. — Discount  is  always  required  by  a  bank 
in  advance ;  thus,  in  having  a  note  for  $350  dis- 
counted as  above,  the  sum  received  would  be  the 
answer  to  tliat  example. 

Banks  aUow  a  time  of  notification  for  payment  of  Day*  of  grac«. 
a  discounted  note,  of  3  days,  called  daj'S  of  grace;  in 
the  discount,  these  are  reckoned. 

6.  What  is  the  bank  discount  of  a  note  of  $1000, 
payable  in  GO  days,  at  6  per  cent,  interest  ? 

Ans.  $10.50. 

7.  My  factor  having  sold  for  me  30  bags  Sea  Island 
cotton,  received  'a  note  from  the  buj-er  for  $3456,  at 
90d. ;  this  note  bQing  discounted  by  the  Charleston 
Bank,  at  7  per  cent.,  what  amount  was  credited  to 
my  account?  ^?is.  $3393.50. 

8.  What  is  the  present  value  of  a  note  of  $450, 
payable  in  90d.  at  the  bank,  at  7  per  cent.  ? 


COMMISSION. 

262.  Commission  ifi  a,  certain  per  centage  charged  commis!>ioB 
by  a  factor,  broker,  or  general  agent  for  the  transac-  <iefinefi. 
tion  of  mercantile  operations. 

263.  The  commission  of  a  factor  is  usuallj'"  forthe  For  what.  s»ie» 
sale  of  cotton,  rice,  and  other  agricultural  products ; [gp^,?}"'"'''*'''" 
of  a  broker,  for  tlie  sale  of  merchandize,  stocks,  un- 

current  money,  bills  of  exchange,  etc. ;  of  a  general 
agent,  for  the  disposal  of  various  kinds  or  a  particular 
class  of  articles. 

264.  The  commission  is  usually  at  the  rate  of  2J  The  usiirI  rn;e 
per  cent.     Occasionally  it  is  more  or  less. 


140 


COMMISSION. 


'To  find  the  an 
swer.  . 


Misitiplyinp 
by  14       ■ 


dtoek  at  par 
Talue; 

Above  par  -, 
Below  par; 
Market  value. 


Example  1. — A  factor  sells  25  bales  of  cotton  at 
$100  per  bale  :  what  is  his  commission  at  2^  per  cent.  ? 

Ans.  $62.50. 
oPKRATiopf,  Eemark. — The  answer 

25x100=2500  is  found  by  multiplying 

2-^          the    amount    bought   or 

sold  by   the  number  ex- 

5000  pressing  the  per  centage. 

Mult,  by  ^     1250  To  multiply   by  ^  is  to 

take    ^    the    amount    of 

62.50  multiplicand.    • 

2.  A  factor  sells  3000  bushels  of  rough  rice,  at  90 
cents  per  bushel :  what  is  his  commission  at  2^  per 
cent.  ?  Ans.  $67.50. 

3.  A  broker  has  $9280  to  be  invested  in  the  Charles- 
ton and  Savannah  Railroad  Stocks,  which  are  15  per 
cent,  above  par,  or  equal  value;  the  broker  is  to  receive 
1  per  cent. :  how  many  shares  of  $100  each  can  be 
purchased  ;  and  what  is  the  commission  ? 


OPERATION. 

.16)9280 


EXPLANATION. 

•  The  premium  being  15 
per  cent,  and  the  com- 
$8000.  quotient,  or  mission  1  per  cent.,  each 
80  shares.  dollar  of  par  value  costs 
$1,  besides  the  premium  and  commission=$1.16. 
Hence,  dividing  9280  by  this,  we  find  the  amount 
purchased  will  be  as  many  dollars  as  the  larger  sum 
contains  the  less. 

When  stock  is  rated  at  its  original  cost,'it  is  said 
to  be  at  its  par  value  ;  when  it  is  rated  higher,  it  is 
said  to  be  above  par,  and  is  at  a  premium;  when 
lower,  below  par,  and  at  a  discount.  Its  commercial 
or  transferable  worth  is  known  as  its  market  value. 

4.  A  broker  invests  for  a  person  $5000  in  certain 
stocks,  which  are  10  per  cent,  below  par,  and  charges 
2  per  cent,  commission  :  what  is  the  sum  invested  ? 

Ans.  $5434.78. 

Ifote. — As  the  stock  is  at  a  discount,  the  $5000  must 
be  divided  by  90+2=92. 

5.  A  commission  merchant  has  sold  for  a  Southern 
factory  25  bales  of  kerseys,  for  $2550 :  what  is  the 
amount  of  his  commission,  at  2^  jDcr  cent. ;  and  what 
is  the  amount  to  be  j)aid  the  owners  ?  ' 


SIMPLE   FELLOWSHIP.  141 

6.  An  auctioneer  sells  at  public  sale  20  casks  bacon, 
at  an  average  pHce  of  $56  per  cask  :  what  is,  his 
commission  at  2^  per  cent,  and  what  is  due  his 
employer  ? 


SIMPLE  FELLOWSHIP. 

265.  Fellowship  or  Partnership  is  an  association  or  Fellowship  <ic- 
connection  formed  for  commercial  or  trading  purj)Oses.  ^"^^^^ 

266.  The   money  invested   by  the   individuals   so^j^^  ^    j,^,  j^. 
associated  or  connected  is,  commercially,  the  stock  vested. 

or  capital. 

267.  The  gain  resulting  is  called  the  profits;  the  The  profits; 
decreased  value  of  the  capital,  through  failures  or 
various  mercantile  casualties,  is  known  as  the  loss.      "^^^  ^^'^' 

26S.  The  profits  shflred  by  the  persons  connected  rrofit  divi- 
are  called  profit  dividends;  the  losses,  loss  dividends.  Loss  dividends 

Example. 1. — Two  persons*  form  a  business  con- 
nection ;  one  invests  $500,  the  other  $700,  and  they 
gain  $480  :  how  is  the  gain  to  be  divided  ? 

Remark. — Since  one  invests  -iV>  lie  is  to  have  -^The  method  to 
of  the  gain,  which  is  $480=$200;  and,  similarly,  thegaTorYoIr"' 
other  is  to  have  -iV=$280.     We,  simply  to  find  loss 
or  gain,  multiply  the  w'hole  gain  or  loiAs  by  each  of 
the   person's   or   company's   fractional   part   of  the 
stock. 

3.  Three  merchants  form  a  partnership ;  the  first 
furnishes    $3000,   the   second   $5000,  and    the   third 
$7000 ;    their  gain  was   $3000 :    w^hat  were  the  divi-     , 
donds?  ^ns.  The  Ist.,  $G00;  2d.,  $1000 :  3d.,  $1400. 

3.  Three  merchants,  who  had  severallv  furnished 
$3000,  $5000,  and  §7000,  lost  by  a  failure  $G00  :  what 
part  of  the  lors  belongs  to  each,  and  what  amount? 

-'1"«>^-  hi,-^-;  $120,  $200,  $280. 

4.  A,  B  and  C  freiglit  1  a  ship  Avith  270  tons.  A's 
shipment  was  IXi  tons,  B's  72,  C's  102.  In  a  storm, 
90  tons  were  thrown  overboard:  what  was  the  loss 
on  1  ton,  and  how  lu  .ny  tons  did  each  lose  ? 

Ans,  i  loss  on  each  ton,  and  A's  loss  32,  B's  24,  C's  34. 


142  SIMPLE   PELLOWSBIP. 

5.  A  ship,  Valued  at  $25,200,  was  lost  at  sea ;  there 
Was  an  insurance  on  her  of  $18,000  :  what  was  the 
loss  to  the  owner  A,  whose  investment  was  to  the 
value  of  i ;  to  B,  whose  was  i^ ;  to  C,  whose  was  i  ? 

^«s.  A,  $2400;  B,  $3600^  C,  $1200. 

6.  A  person,  failing  in  business,  owes  A,  $350 ;  B^ 
$1000  ;  C,  $1200;  D,  $420  ;  E,  $85  ;  F,  $40  ;  G,  $20; 
he  has,  to  nieet  these  amounts,  $1557.50 :  what  will 
be  each  creditor's  share  ? 

7.  If  a  town  raise  a  tax  of  $1920,  and  all  the 
property  be  valued  at  $64,000,  what  will  it  be  on  $1  ? 
what  will  a  person's  tax  be,  whose  j)roperty  is  ap- 
praised at  $1200  ?  Ans.  .03  on  a  dollar;  $38. 

Assessment  of      When  taxcs  are  assessed,  an  inventory  of  all  real 
*"*'■  and  personal   property  of  the  whole  town  is  first 

Capitation  tax.taken;  /and  when  there  is  the  capitation  or  poll  tax, 
that  of' each  one  subject  to  it  is  ]Dut  down.  As  the 
polls  are  specifically  rated,  that  tax  is  to  be  first  sub- 
tracted from  the  whole  tax,  and  the  remainder  to  be 
assessed  on  the  property.  To  find  the  amount,  each 
individual  is  to  be  taxed  for  his  property,  We  find 
,  how  much  the  remaiilder  of  the  whole  tax  is  on  $1, 
and  multiply  his  inventory  by  it. 

General  tax.  In  some  States  there  is  no  capitation  tax,  and  the 
sum  to  be  raised  for  the  expenses  of  the  Government 
is  collected  from  each  individual,  in  proportion  to  his 
property.  In  South  Carolina,  this  is  on  "land  and 
negroes,  and  is  called  the  general  tax.     Apart  from 

spe(SRi  tax  this,  there  are,  in  incorporated  towns,  special  taxes 
on  houses,  serx^ants,  carriages,  horses,  etc. 

8  In  a  certain  town  where  there  is  the  capitation 
tax,  the  amount  to  be  raised  is  $5999.  The  real 
estate  is  valued  at  $500,000  and  the  personal  at 
$300,000.  There  are  666  taxable  polls,  each  of  which 
•  is  assessed  $1.50  :  what  is  the  tax  of  A,  wjiose  real 
estate  is  valued  at  $4000  and  his  personal  property 
at  $8000,  and  who  j)ays  one  capitation  tax  ? 

Ans..  $76.50. 


DOtJBLE    FELLOWSHir'.  143 


DOUBLE   FELLOWSHIP. 

269.  Double  Fellowship  varies  from  Simple  Fellow- DouMe   felio-*- 
ship  in  the  iuvestment  of  shares  in  a  company  for^^'P  defined, 
unequal  terms  of  capital  and  time. 

Example  1. — A  and  B  form  a  partnership  for  the 

transaction  of  certain  business.     A  puts  in  8300  for 

8,  and  B  $400  for  7m.;  their  gain  is  $156  :  what  is  the 

shafe  of  each  ?  A's  ^300  for  8m.=2400  for  Im.  , 

B's  8400  for  7m.=2800  for  Im. 


5200  for  Im. 
Hero  We  have  the  joint  stock  of  $5200  for  1  month,' 
of  which  A  j)uls  in  §2400  and  B  $2800  ;  thus,  A  is  to 
have  -iti^=-i\  of  the  gain,  and  B  ^1=1^;  "r,  A  fj 
of  $156=^72,  aud  B  -^  of  $156=$84. 

li^otr. — Each  partner's  stock  is  mfiltiplied  by  the 
time  of  its  engagement. 

2.  Three  merchants,  A,  B  and  C,  enter  into  part- 
nership ',  A  puts  in  $60  for  4m.,  B  $50  for  10m.,  anJ 
0  $80  for  12m. ;  they  lose  $50  :  what  is  the  share  of 
loss  to  each  ?     Ans.  A,  87.05;  B,  $14.70  ;  C,  $28.23. 

3.  Four  traders  form  a  connection  ;  A  puts  in  §400 
for  5m. ;  B,  $600  for  7m. ;  C,  §900  for  8m. ;  D,  §1200 
for  9m ;  they  lost  §750  :  what  loss  did  each  sustain  ? 

Ans.  A,  §60.77 ;  B,  $127.63  ;  C,  §233.38  ;  D,  328.20. 

4.  A  commenced  business  November'  1,  with  a 
capital  of  $3400.  On  the  Ist  February  he  associated 
with  him  B,  who  invested  his  capital  of  $2600.  In  a 
twelvemonths'  time,  they  had  gained  §750:  what  i» 
the  share  of  each  ?   A7is.  A's,  $476.63  ;    B's,  §273.36. 

5.  Two  merchants  having  entered  into  partnership 
for  16m.,  invested  as  follows :  A,  at  tirst,  to  the 
amount  of  $1600,  and,  at  the  end  of  9m ,  $900  more; 
B,  at  first,  $650,  and,  at  the  end  of  6m.,  withdrew 
1350;  they  gained  §500:  what  was  each  one's  part? 

6.  On  the  1st  January,  A  commenced  business  with 
$940  ;  on  the  Ist  February  after,  he  associated  with 
him  B,  who  put  in  §660 ;  on  the  Isfc  June,  C  was  ad- 
mitted to  the  firm,  with  a  capital  of  $1800;  at  the 
end  of  the  year,  they  had  gained  §992  :  what  was 
the  share  of  each  ? 


144  INSURANCE. 


INSUKANCE. 

» 

Insurance  de-      270.  Insurance  is  the  engagement  of  a  company  to 
fined.  protect,  for  a  specified  time,  a  certain  property  from 

loss  by  fire  or  other  casualty. 
The  policy.  2T1.   The  Written  contract  assuring  such  protection 

is  called  the  policy. 
The  underwri-      272.  The  persons  pledged  to  its  performance  are 
ters.  known  as  underwriters. 

The  premium.       273.  The  sum  paid  for  such  risk  or  service  is  termed 

the  premium. 
Fire,  marine,        274.   Fire,  marine  and  life  insurance  embrace  the 

and  hie   insii-      .   ,       „  ,  .'■,,.    .  .     ' 

ranca  cover  all  riSKS  lor  which  policics  are  givcn, 
us^per  centage     275.    The  per  centage  for  which  insurance  against 
varioua.  fire  18  effected,  varies  according  to  the  nature  of  the 

property  or  its  locality ;  against  marine  or  sea  disas- 
ters, according  to  the  strength  of  the  vessel,  the 
voyage,  or  other  circumstances  in  such  connection  j 
against  the  loss  of  life,  according  to  the  age,  health, 
or  exposure  to  sickness  or  danger  of  the  individual. 
Example  1. — What  would  be  the  premium  for  the 
insurance  of  a  house  valued  at  $5000,  against  loss  by 
fire,  for  lyr.,  at  |  per  cent.  ?      (5000-r-2=25.) 

Ans.  $25. 

Note. — We  simply  divide  by  the  denominator  of 
the  fractional  per  centage,  ^,  and  as  it  is  a  per  cent, 
diviisor,  pont  off  two  figures  in  the  result  for  cents. 

'  2.  What  would  be  the  premium  for  insuring  a  ship 
and  cargo  valued  at  $37,500,  from  Charleston  to 
Liverpool,  at  3^  per  cent.  ?  Ans.  $1312.50. 

OPKRATION. 

37500 
3^ 


112500 
18750 

1^2.50 


KXPLANATIOK. 

We  multiply  by  3,  and 
to  its  result  add  the  re- 
sult of  the  multiplication 
by  ^  (37500-^2=18750). 


3.  What  is  the  insurance  on  a  store  and  goods 
valued  at  $15,000,  at  2i  per  cent.?        Ans.  $337.50. 


rROFIT   AND   LOSS. 


•4.  A  merchant  owns  5  of  a  ship,  valued  at  $24,000, 
5ind  insures  his  part  at  2V  percent.:  v,]int  does  he 
pay  for  the  policy  ?  '      .  A/^s.  $450. 

5.  A  shipment  of  goods  from  Liverpool,  valued  at 
-C.334  10s.  Gd.,  has  been  insured  at  IJ  per  cent. :  in- 
cluding cost  of  policy  tiaper,  $1.25,  what  must  be 
paid  in  American  money,  ealcnlatinj^  the  pound  at 
^4.87  ?  [  "  Ans.  646.80. 

.  6.  I  have  effected  an  ihsurauce  on  'a  friend's  house 
in  Augusta,  Ga.,  for  $8000,  at  I  percent.;  furniture 
$1500,  at  1\V  per  cent.  ;1  stable,  hor.scs  and  cai-i-iage 
$2000,  at  3  per  cent. :  what  is  the  insurance  i* 

Yl    ,  ,      _  /  .  Ans.  ii;108.75. 

7.  A  yessel  and  cargo!  worth  $65,000,  are  damaged 
to  the  amount  of  20  per  cent.,  and  there  is  an  insui->- 
nnce  of  50  per  cent,  on'  the  lo/>s  :  how  mucR  will  the 
owner  receive  ?  Ans.  $6500. 

8.  -What  will   l>e  the  annual  premium  for  insuring 
'$5000,  for  12  years,  the  iifv-  of  a  man  40  years  old,  at 

the  premium  of  fl  ('('  p;':-  c-nt.  ? 


■PilOllT  AND.  LO^-6. 

STG.   /  V-;//i  and  J.oss  is  the  name  applied  to  the  procor;?  pioflt  j 
tised  in  mercant^e  transactions  to  ascertain  the  favorable *^'''^'"''' 
-or  unfavorable  result  of  a  financial  operation.     It  is  a  eoai- 
bination  of  rules  already  known,  and  very  important  in 
this  connection  as  showing  a  method  practically  valuable 
to  assure  a  certain  per  cciitage  of  profit  or  loss. 

Example  1. — Bought  800  lbs.  of  sugar  a*  9  cents  per 
lb.,  and  sold  the  sarac  for  12*  cents  per  lb. :  what  was  the 

Am.  $10.50. 

EXPLANATION. 

Here,  after  the  multipli- 
cations of  purchase  and  sale, 
the  difference  is  found  ho- 
twccn  t'-  -  '"- »  for  answer. 


profit  ?, 

OPE 

RATIO.?;. 

I^urchase. 

^Sa^e. 

800 

300 

9 

m 

27.00 

:)600 

150 
:i7.50 
27.00 
10.50 


H 


146 


PROFIT    AND   LOSS. 


2.  Bought  250  yds.  of  cloth  atl;225.  per  yd.!:  what  musi 
it  be  sold  for  to  gain  7  per  ceni.  ?  '  Am.  $601.87, 

EXPLANA'ilON.  . 

HaviDg  found  the  cost,  it 
is  neccF^ary  to  Multiply  that 
amount  by  th<;  gain  pro- 
posed, and  we  find  the  an- 
swer $39.37;  acidiug  this  ti> 
the  cost,  we  obtain  for  an- 
swer,as  above.    ' 


OPIORATIOK. 

250 
225 


1250 
500      . 
fOO 
'"       S5G2.5G=tlie  cost. 

7  gain  proposed. 
~W9.37V>0  .  I 

$562^ 
•,     ,        -SG01.S7 
Kepetitior..  of        Xote. — In  the  multiplication  of  dollars  and  ceats  by 
poinf off  cemin  cents,  let  it  be  remembered  that  4  figures,  uiust  be  pointed, 
figures.  .oft".     The  two  at  the  extreme  right  are  not  in  these  calcu--' 

lations  usually  considered.     The  next  two  arc  cents  ;  those 
at  the  left  of  the  point  are  dollars. 

3.  Bought  14  bbl.  flour  for  $100,  and  sold  the  same  at 
$8  per  bbl. :  what  was  the  gain  ?  I  Ans.  $12. 

4.  Bought  100  bushels  corn  for  75  cents  per  bushel/and 
sold  it  for'$69  :  what  was  the  loss  ?  . 

5.  Bought  58  bushels  sweet  potatoes  at  62^-  per  bushel,/ 
and  sold  them  at  75  cents  per  bushel :  what  was  the  gain  " 

6.  Bought  60  hhds.  of  wine  at  $39  Jhhd.,  and  sold  tlj 
same  at  the  rate  of  $50  a  hhd.,  but  disc-OfUnted  3  per  ceul 
for  cash  :  what  was  the  profit  ?  ^ 

7.  Bought  325  bbls.  of  flour,  at  ^5.35  per  bbl.,  and  sold 
them  at  $6.25  per  bbl.,  with  a  discount  of  2  per  cent,  for 
cash  :  what  was  the  profit  ?  \ 

8.  Bought  7  hhds.  of  bacon  for  §4ii0,  and  sold  them  ih[. 
$500  :  what  was  the  gain  per  cent.,  ok  per  100  ? 

Ans.  S8-H-. 


Note. — This  can  be  solved  by  simple  proportion,  and  is 
the  same  as  if  the  question  had  been  thus  put :  If  $460.00 
gain  $40,  what  will  $100  gain"? 
460:40:100: 

40=the  diffocence- between  460 
M         [and  500. 


460)4000(Ahs-  $8if 


3680 
Eeducedby    10)~820 


460  or  ff,  <)T  ii 


PROFIT   AND    LOSS.  147 

9.  Bought  4hhcls.  of  sugar,  each  weighing  2501bs.,  for 
$70.00,  and  sold  them  again  at  8  cents  per  lb. :  how  much 
was  the  profit  per  cent.  ? 

Note. — Find  the  net  profit  on  the  whote,  then  the  per 
cent,  by  proportion,  as  in  the  8th  example.    ' 

10.  What  is  the  profit,  per  cent.,  on  a  yard  of  cloth, 
bought  at  $4  and  sold  at  $5  ? 

Cost  :  Gain  ::  Par  value  :  Gain  per  cent.- 
$4    :    $1     ::  lOOpcrct.  :  25  per  cent.     Am. 

Note. — The  par  value  of  an  article  is  its  first  cost.  Par  vaiu.>  a-v 

fiuc'd. 

Remark. — Here,  we  ascertain  the  differepce  between  a  propoit.om^i 
the  purchase,  price  and  sain,  and  say,  as  the  purchase  statement. 
price,  or  cost,  is  to  the  total  gain,  or  losa^^so  is  100  per 
cent.,  or  par  value,  to  the  gain  or  loss  per  cent. 

11.  Bought  500bbl.  pork  at  §12  per  bbl.,  and  gave 
note  at  6m.,  in  settlement :  v/hat  was  the  profit,  the  same 
having  been  sold  at  $15  per  bbl.,  cash  ? 

12.  A  merchant  purchased  20  chests  of  green  tea  at 
$40  per  chest,  and  settled  for  them  with  a  note  payable  in 
Cm.  He  afterward  sold  them  for  $46  per  chest,-  at  30 
days :  what  did  he  gain,  if  the  bank  discount  were 
reckoned  at  7  per ,  cent.  ? 

Remark.— Find  the  cash  cost  of  the  20  chests,  by 
deducting  3|-  per  cent.  (7-4-2=32-)  from  their  cost,  at  $40. 
Find,  too,  the  cas'li  price  of  sale  by  deducting  ^  per  cent. 
(12-r-7=-i^  or  Im.  int.)  from  the  price  of  20  chestsX$40. 
The  differance  between  the  cash  cost  and  the  cash  price  of 
sale  is  the  profit. 

13.  A  merchant  bought  25hhd.  of  molasses  at  $30  per 
hhd.,  and  gave  his  note  at  90  days.  A  mouth  later,  he 
sold  them  at  ^o5  per  hhd.,  and  received  a  note' payable  in 
Cm.  :  what  was  tlie  profit,  the  bank  discount  being  at  the 
rate  of  6  per  cent.  ? 

Remark. — Here,  2  per  cent,  is  to  be  deducted  from 
the  price  of  the  sale,  as  the  good;^  Were  1  mouth  on  hand, 
and  were  not  paid  for  until  6m.  had  expired;  making  in 
all  7m.  interest,  at  G  per  ctat.,  which,  for7m.=  3i  percent.- 


148 


PROFIT  AND  LOSS. 


The  purchase 

Erice    given  to 
nd  selling 
price. 


OPERATION. 
700 

25 


3500 
1400 


14.  A  grocer  bought  75bbl.  of  cider  at  S2.25  per  barrel; 
and  gave  his  note,  due  in  6m.  Two  months  later,  he  sold 
them  at  $3  a  -barrel,  on  a  credit  of  4m. :  what  was  his 
profit,  the  bank  discount  being  at  the  rate  of  6  per  cent.  ? 

15.  Bought,  an  invoice  of  books  for  $700  :  how  m\ich 
must  the  same  be  sold  for,  in  order  to  gain  25  per  cent.  ? 

Ans.  $875. 

EXPLANATION. 

We  here  multiply  the 
purchase  price  by  the  per 
cent.,  and  add  .the  result 
to  the  cost.  The  amount 
would  have  been  subtracted 
had   the  per  centage  been 

175.00  loss. 

700. 

^875. 

16.  What  must  15  pipes  of  wine  be  sold  for,  which  cost 
$55  a  pipe,  to  gain  12  per  cent,  on  the  cost  ? 

17.  What  must  35  bags  of  cotton  bring,  that  cost  $75  a 
bag,  to  gain  15  per  cent.  ?  * 

18.  Bought  an  invoice  of  figs  at  12|-  cents  per  lb.  : 
proving  injured,  what  must  they  be  sold  for  to  lose  10  per 
cent.  ?  Ans.  11  cents  and  2i  mills. 


A  proposition  in 
profit  and  loss 
solved. 


Note.— This  is  easily  solved  by  proportion  ;  thus, 

OPERATION. 

100  :  90  ::  12^ 
90 


1080 
45 


100)1125  (11}  or  lie.  2J  mills, 
100 


125 

100 


Reduce  by  25)  YW=i 


EXPLANATION. 

Here,  we  say,  as  100  is  to 
90  (=100—10)  so  is  the 
purchase  price  to  the  sell- 
ing price. 


19.  A  trader  sold  apples  at  $1.50  per  bbl.,  and  lost  10 
per  cent,  by  the  sale  :  what  was  the  cost  ?     Ans.  $1.66-J-- 


EQUATION    Oi-    PAYMENTS.  1-19 

OPKRATION. 

90  :  100  ::  150  : 
100 


90)15000(lG6f 
90 


600 
540 


GOO 
510 

60 
20.  A  merchant  bought  a  piece  of  velvet  at  85  per  yard, 
but  it  being  daxiiaged,  he  proposed  to  sell  it,  so  as  to  lose 
20  per  ceat. :  how  must  it  be  marjced,  so  that.  10  per  cent, 
be  deducted  from  tlie  price  at  wliich  he  hud  designed  to 
sell  it?  90  :80  ::  5  :  Ars. 

Remark. — Here,  it  is  said,  as  100,  diminished  by  the 
per  cent,  to  be  deducted,  is  to  100,  increased  by  the  per 
cent,  to  be  gained,  or  diminished  by  the  per  cent,  to  be 
lost,  so  is  the  cost  to  the  proposed  price  of  sale.' 

'21.  Bought  a  mule  for  $175  :  what,  so  as  to  lose  5  per 
cent,  on  the  cost,  .shall  I  ask  for  it,  so  that  I  may  diminish 
the  price  20  per  cent..? 


EQUATION   OF   PAYMENTS. 

217.  E(j:Hatwn  of  PaipnenU  is  an  operation  to  discover  E.iuation  of 
the  mean  time  for  settlement  of  debts  incurred  at   pre-  payments  d<>. 
vious  dates,  so  that  neither  party  interested   shall  suffer 
loss. 

ExAMPLK  1. — Supposing  a  person  owes  me  $100,  due 
in  30  days;  8200, .due  in  25  days;  and  $150,  due  in  10 
days,  but  wishes  to  settle  all  at  once :  what  is  the  mean 
time  of  payment ;  or,  enuivalentl}'.  how  long  is  he  to  keep 
the  money  if  he  wishes  to  pay  all  at  once  ? 


150 


ECJUATION    OF  PAYMENTS. 
OPERATION. 

aiOO  $200  $150 

30  25  10 


3000  1000  1500 

400 


5000 
These  added  =9500,  which,  divided  by  450,  the  amount 
of  the  different"  debits,  gives  21  nearly,  and  is  the  time 
sought. 

To^  equate  pay-  Remark. — From  this,  we  lenrn  to  multiply  each  debt 
by  the  time  at  ^vhicu  it  falls  due,  and  divide  the  product 
by  the  indebtednesses,  for  the  mean  titne. 

2.  I  owe  $500,  one  half  payable  to-day  and  the  remainder 
in  8m.  :  when  is  the  equated  day  for  settlement  ? 

A7i$.  4m. 

3.  A  owes  B  three  different  sums  of  money — $150  due 
in  3m.,  S240  in  60  days,  and  |300  in  4m. :  what  is  the 
mean  time  of  payment  ? 

4.  A  owes  B  $500  due  in  3m.,  $350  in  2m.,  and  $200 
in  Im. :  what  is  the  mean  time  of  payment  ? 

5.  A  owes  B  four  different  sums  of  money,  as  follows: 
$1000  due  in  3m.,  $500  in  6m.,  $400  in  5m.,  and  $300  in 
2m. :  as  he  wishes  to  settle  these  at  one  time,  what  is  the 
equated  date .? 

6.  A  owes  B  $400,  which  is,  by  agreement,  to  be  paid 
in  4  equal  instalments ;  the  first  in  30  days ;  the  second 
in  60  days;  the  third  in  90  days,  and  the  fourth  in  4m. 
He  wishes  to  settle  all  at  one  date  :  what  is  the  equated 
time  ? 

7.  A  merchant  purchased  a  certain  commodity  for  which 
he  gftve  his  notes ;  one  of  $300,  payable  in  3m.,  and 
another  of  $350,  pa3'able  in  6ni.  One  month  after  the 
purchase,  he  proposes  to  give  a  single  note  for  the  whole 
amount :  for  what  time  must  the  note  be  written  ? 

Remark. — The  first  note  having  2m.  of  unexpired 
time,  and  the  second  5m.,  multiply  $300  by  60d.,  and 
$350  by  150d.  (5m.) 

8.  When  is  the  mean  time  for  payment  of  $400  due  in 
6m.,  $500  in  8m.,  and  $1000  in  12m.  ?  Ans.  9-^. 


I  I  •  ■   • 

9.  Two  mcrchanrs  had  the  following  bus;nc?f?  tiansac-- 
tious :  A  [lurehased  of  ]5  a  bill  of  goods,  I^Iay  In,  1860, 
on  Sra.  crodjf,  for  6200 ;  May  1,  1860,  on  4ni.  credit,  for 
$600 ;  and  B  purchased  of  A  a  bill  of  goods,  May  15, 
1860,  on  3m.  credit,  for  ^oOO;  June  M,  l^<60,  on  4m. 
credit,  for  $900  :  when  does  the  balance  owned  by  B  fall 
due?  \    '^      1 

As  A's  dept  is  found  to  be  due  August  20,  and  TVs 
8ept(imber  20,  40  days  after  A's  is  due,  the  question  i^ 
when  is  B  to  pa}'  the  balance  of  $400  ?  \ 

Remark. — Had  A  and  B  each  paid  their  'debts  when 
the  time  was  up,  the  question  would  have  been  plain  ;  but 
as  the  account  is  to  be  settled  by  B'«  paying  the  balance 
of  S400,  he  can  keep  that  sum  long  enough,  after  Septem- 
ber 29,  to  equal  A's  holding  8800  40  days  after  it  was 
due;  as  8800  for  40  days  equals  S400  for  80  days=32,000, 
and  this,  divided  by  400=80.  Hence,  80  days  from 
September  29,  rs  the  mean  time. 

10.  Two  houses. in  Savannah  had  the  fullowing  transac- 
tions:  A  purchased  a  bill  of  B,  January  7,  1861,  Im, 
credit,  for  §800;  February  7,  1861,  2m.  credit],  for 
S566.66^;  February  7,  1861,  3m.  credit,  for  ^433.33J. 
B  purchased  of  A  a  bill.  January  18,  1861. '2m.  credit,  for 
S200  ;  February  26,  1861,  4m.  credit,  for  §1200  ;  March 
1,  1861,  3m.  credit,  for  $800  ;  March  26,  1861,  3m.  credit, 
for  $800  :  when  shall  B  pay  to  A  the  balance  ? 

Ans.  January  27,  1862. 


CABTT^R. 


'278.  Barfcrrin  the  excliange  of  commercial  values.  Bartov  define 

^79.  Though\  separately  placed,  it  rightfully  belongs  to  Its  qucsiion=» 

thV  solution  of  (\ucstions  by  analysis.  '  are  tlioso  oi 

J'jXAMPLE  1. — How  many  pounds  of  butter,  at  22  cents 

per  lb.,  must  be  given  for  a  chest  of  tea,  containing  751bs., 

;>t  80  cputs  per  lb.  ? 


PRACTICE. 


OPERATION. 

75 

80      ' 

22)v^0.OO(272if=-,V 
4-1 

IGO 
154 


EXTLANATIOS'. 

The  value  of  the  tea 
being  ascertained,  the  quan- 
tity of  ];»utter  to  pay  for  it 
is  found  by  dividing  the 
tea's  value  by  the  price  per 
lb.  of  the  butter. 


GO 
.44 

lo 

2.  A  has  300yd.  of  cotton  bagging  worth  30  cents  per 
yard,  which  he  wishes  to  exchange  for  corn  at  75  ceat« 
per  bushel :  how  many,  bushels  can  he  have  ?  Ans.  120. 

3.  A  flour  merchant  had  200bl)l.  of  fiour,  valued  at 
is;  10.50  per  barrel,  for  which  a  grocer  gave  him,  in  money, 
$1090,  and  the  balance  in  Cuba  molasses,  at  20  cents  pcv 
gallon :  how  many  hhd.  of  molasses  did  he  receive  ? 

-i.  What  number  of  barrels  of  apples,  at  $1.20  per 
barrel,  will  purchase  20  cords  of  wood,  at  §3.50  per  cord  f 

5.  A  merchant  exchanged  1  case,  50pcs.=  1500)al. 
calico,  worth  9  cents  per  yd.,  for  G0pcs.= 1800yd.  long 
cloth,  at  10 J  cents  per  yd.:  what  was  the  difierence  that 
he  had  to  pay  ? 

6.  A  trader  received  in  exchange  for  300pr.  brogans, 
valued  at  95  cents  per  pair,  CO  hides,  at  $l.G2i-  each  :'  how 
much  was  the  balance  in  his  favor? 


pbactiok! 

280.  /.'.(c/<.  i,-  a  process  for  solving  questions,  by  sub  • 
stituting  for  large  multiplications  and  divisions  aliquot 
parts,  such  as  l,  i,  1,  etc,  -        • 

.i.trrtotjono'"     281.  It  is  mainly  a  contracted  form  of  rules  alreadr 
'■'■  ''^'"'-       considered,  but   especially  of  proportion.     A.8   its   namfe 
indicates,  the  method  of  solving  questions  proper  to  i\ 
can  only  be  acquired  by  constant  use. 


PRACTICE. 


1  :>:; 


OrERATION. 

\  clollar)48 

i  dollar  i8=i  of  k  -^4 
.12 


Example  1. — Wliat  is  the  cost  of  48yd.  of  satinet,  at 
75  ceuts  per  yd.  ? 

EXl'LAXATION. 

We  here  first  find,  by 
dividing  the  yds.  by  i  dol- 
lar, the  value  at  J  dollar  a 
yard ;  and  then,  taking  I 
of  that  answer,  and  adding 
S36^1?w.  it  to  the  sum  of  the  yd.  at 
4-Hlollar,  find  the  true  result.    (J+.l  =  t  and  \  dollav=.75. ) 

2.  ^Vhat  is  the  value  of  80yd,  of  furniture  chintz,  at  20 
eonts  a  yard  ?     {^  dn}hir=20  cents.) 

3.  What  is  the  price  of  36  barrels  of  ale,  at  $3.75  per 
barrel ? 

EXPLANATION. 

We  here  first  multiy»ly 
by  o  for  the  value,  at  $3 ; 
then,  we  take  J  of  36,  for 
the  50  cents'  value;  and. 
kst,  \  of  18,  whicrh'  is  ^ 
the  value  of  50  cents. 

$135 

4.  V/hat  is  the  value  of  42yd.  of  woolion,  at  25   cent'* 
pci-  yd.  'i 

5.  What  will  5cwt.  8qr.  161b,  cost.,  at  $4.20  per  cwt.  ? 

An^.  S24.82; 

6.  What   is   the  price  of  60yd.  of  ladies'  cloth,  at  62^ 
ceuts  per  yd. '( 

7.  Wh:it  will  250yd.  of  muslin  cost,  at  372   cents  per 
yPiTd  ? 

8.  What  vrill  130vd.  cassimere  cost,  at  87  J   cents  per 
yjlrd  ? 

9.  What  l>s  the  value  of  160yd,  of  cotton  flannel,  at  12^ 
ceuts  per  yd.  ? 

10.  What  i&  the  value  of  140yd.  of  broadcloth,  at  £1 
12s.  6d.  per  yd.  ? 


Ol'EU 

ATIOX. 

36 

3 

11)8 

of  3f 

5=   18 

of  1 

^=     0 

OPEIIATION, 

£,       S. 

Fov  £1  =140  00 

For  10s.  (•!  £)  70  00 
For  2s.  (i  10s.)  14  00 
For  6d.  (1  2.S.)    3  10 


,\m,   £227  10 


EXTLANATION, 

At  £1  per  yd.,  140yd.= 
£140.  As  10  shilliugs= 
£i,  V\c  take  for  the  10s.  ^ 
£140=£70.  As  2  shillings 
=\  10  shillings,  we  take  \ 
£70=£14,  which  is  the 
value  ibr  10  shillings.     As. 


the  6d.=^  of  1,  or  I  of  2  shillings,  we  fake  i  £U—£3 
10s.     These  values  added  give  the  result  sought. 

11.  What  will  36  bushels  of  wheat  cost,  at  7s.  6d.  per 
bushel?  ■  yl«s.  £13  10s. 

12.  What  is  the  cost  of  3cwt.,  3qrs.  211bs.,  at  £3  15s. 
Gd.  per  cwt.  ? 

OPKRATION. 

£3     15s.    Gd.      . 
3 


11 

G 

G     cost  of  3 cwt. 

^ 

OCWt.       =    1 

17 

0     cost  of  2qr3. 

i 

2qrs.      = 

18 

10^  cost  of  Iqr. 

h 

Iqr.       = 

9 

b\  cost  of  I2|lbs.* 

i 

12ilbs.= 

4 

H  cost  of  Gi  lbs. 

An^,  £U     17s.     3^. 
13.  What  will  18yds.  cloth  cost,  at  14s.  Gd.  per  yd.  ? 
U.  What  will  24y(ls.  cloth  cost,  at  12s.  per  yd.  ? 

15.  What  will  2cwt.,  2qrs.  18lbs.  sugar  cost,  at  87.50 
per  cwt.  i* 

16.  What  will  Ggajs.  3qts.  oil  cost,  at  $2.25  per  gal.  ? 


y%  «frecto<1. 


EXCHANGK. 

Rx«iiijp^'<'  (io-       2§2.  Excliange  is  a  convenient  process  for  the  tran?;- 

""*■'''  mission  of  funds  to  persons  residing  or  travelling  abroad; 

or  to  parties  living  in  other  places  of  the  same  country. 

n«»»ex.!ian;.'r  283.  Sucli  transmission  is  effected  through  the  medium 
of  a  draft  or  bill,  which  is  commonly  an  order  from  .1  bank 
to  an  agent  commercially  in  its  connection;  or  from  a 
banker  or  broker,  whose  business  is  the  transfer  of  such 
paper  to  some  house  with  which  he  has  financial  under- 
standing. 

;^'!.J*!fV'in'''"'^  284.  When  the  parties  are  residents  in  the  same  coue- 
f  ry,  such  draft  is  called  an  inland  bill ;  otherwise,  a  foreien 
bill. 


i»rfig»  bill. 


*  1  quarter— .251  bs. 


EXCHANGE. 


155 


285,  The  value  of  the  pound  sterling  (Art.  104),  is  Tim  vaiu«  of 
1*4.44,4  {=S^),  at  par.     But  in  settling  accounts  between  {,';,y'°"°''  ''^'" 
this    country  and   Great  Britain,  or  in   the  purcha.«e  of 
hillf»  of  exchanire,  it  is  seldom  that  Britisli  currency  is  to 
be  had  at  that  rate.     Usually  it  is  at  an  advance  of  5  to  2*0  ^^^e!'""^'"  """" 
per  cent.     Sometimes  it  is  at  a  discount,  or  below  par,  or" 
)>elow  the  intrinsic  value  of  $4.44,4  mills. 

a§6.  A<j:reeablv  to  thase  fluctuations,  exchancre  is  said  Bill? at p»r, pr»- 

,  '^  •'  .  T  i.  niiuta,  or  di(<- 

to  be  at  par,  at  a  premium,  or  at  a  discount.  count. 

Example  1. — A  merchant  in  Charleston  wishes  to  pay 
for  a  bill  of  sugar,  due  in  New  Orleans,  $2550.  He  finds 
that  exchauge  is  at  2  per  cent,  premium  :  what  must  he 
pay  for  a  draft  ?  *         ^7).?.  2G01. 

OrERATIOX. 

•  2550 

9 


$51.00  prom. 
$2550.      bill. 


^2601.      draft. 

2.  A  company  in  Savannah  wish  to  remit  to  Galveston, 
Texas,  $500,  and  find  that  exchange  is  1  per  cent,  below 
par  :  what  must  they  pay  for  their  bill  ?  Aim.  $405. 

3.  A  company  tradinj^  iii  Mississippi,  propo-se  to  remit 
to  Richmond,  Va.,  $5000,  but  find  a  bill  cannot  be  pur- 
chased under  2^  per  ceut.  premium:  what  will  the  per 
centage  be  ? 

4.  A  banker  in  Charleston,  South  Carolina,  wishes  to 
send  to  Liverpool  a  bill  of  exchange  for  £112  15s.  6d. : 
what  must  he  pay  for  the  bill  when  exchange  is  0^  per 
cent,  premium  ?  '  Ans.  £123  4,s. 

OPEEATION.. 

£U215s.6d.=£112.775(Art.  214, 14  Ex.)  add  9  J-  per  ct. 
10.713 


£123.488x40  aud^9=$548.83. 

HKitfARK. — In  bills  of  exchange  oh  England,  the  £  ster- How  the  £  is 
ling  is  still  valued  at  $4i=  84.44,  instea'd  of  its  present  ^*'"'''^- 
worth,  $  1.84 :  hence,  £1=$4.44 
Add  9  per  cent.     .399 


$4.83,9  which  expresses  the  value. 


I5G 


EXCHANGE. 

Note. — Pounds  and  decimals  of  a  pound  are  reduced  as 
above,  to  dollars,  bj  nitilti plying  by  40  and  dividing  by  9.* 

•It  is  convenient  to  be  able  to  exchange,,  expeditiously, 
British  .into  American  currency.  The  following  metlioil 
will  be  found  useful  to  facilitiite  such  computation: 

5.  Change  19.s.  4^d,  to  a  decimal  fraction  of  a  pound. 

EXPLANATION. 

"We   take  half  the  even 


An  easy  way  t(«  Ol'EUATION. 

into  American  ^'^^^  ^^*^' 

»vji.vy.  0.9 


18 

1 


number  of  shillings  for  a 
first  decimal  figure ;  in  this 
example,  9  ;  we  then  change 
the  remaining  pence  and 
farthings  to  farthings,  whicli 
Ans.       0.969  are   the   second    and    third 

detimal  figures;  in  this  example,  18  ;  then,  as  the  number 
of  shillings  was  odd,  we  add  5  in  the  second  place,  and, 
as  the  number  of  farthings  exceed  12,  we  add  1  to  the 
third  figure. 

Xotc. — ]3y  multiplying  decimal  0.969 
b^  40 

and  dividing  by  9)88760 


wc  have  answer  in  $,  cents  and  mills.S4.30,Gm. 

0.  Change  15s.  Gd.  to  a  decimal  fraction  of  a  pound, 

Ans.  0.775=S3.44. 

7.  Change  l.xs.  5d  to  a  decimal  fraction  of  a  pound. 

8.  Change  17s.  3 Ad.  to  a  decimal  fraction  of  a  pound. 
0.  Change  16s.  2>]d.  to  a  decimal  fraction  of  a  pound. 

Ans.  0.89=  83.95,5m. 

OPERATION. 

16s.    2J,d. 

0.8 
9 


0.S9 


Xotc. — In  this  example,  the  shillings  being  even  and 
the  farthings  less  than  12,  the  addition  of  the  5  and  tht 
1  are  not  required. 

10.  Change  10s.  to  a  decimal  fraction  of  a  pound. 

Ans.  0.5=  82.22,2  m. 


EXCHANGE.  157 

11.  Change  16s.  Td.  to  a  decimal  fraction  of  a  pound. 

12.  I  wish  to  remit  to  London  a  bill  of  exchange  for 
£90  :  how  much  must  be  paid  for  the  bill,  when  exchange 
is  at  9 1  per  cent,  premium  ? 

•       Ol'KRATIOX. 

^90 
Add  9^      8.55 


£98.55=  Am.  money,  $438. 

13.  Imported  from  l'vnp:;land  a  bill  of  goods  amounting 
to  £75  6s.  3d.  :  what,  iu  American  nioney,  will  that  be, 
when  exchange  is  5  per  cent.  ? 

14.  My  factors  in  Mobile  shipped  to  Liverpool  55  bale.s 
of  cotton,  weighing  22,500  pounds;  it  was  sold  r.t  Lid. 
( I?.  3d.)  per  lb.  The  freight  and  other  charges  amounlrd 
to  £150  15s.  Having  sold  the  bill  of  exchange,  which 
was  received  in  payment,  at  9.i  per  cent,  premium,  how 
much  would  the  amount  be,  in  American  niouey? 

Jns.  $6724.44. 

15.  I  have  requested  my  factors  to  transmit  to  my 
banker,  in  Paris,  a  bill  of  exchange  for' 4644  francs  : 'at 
the  par  of  exchange,  what  will  be  the  amount  in  Amcricaa 
money  ? 

Rkmark. — French  currency  is  computed  in  francs  andi'renc}:  funm- 
centimes;  the  franc  being  equal  to  18 J  cents,  American *^^' 
money,  and  each  centime  ^-J-y  part  of  a  franc.     To  change  proncb  nx 
francs  to  American   money,  at  the   par  of  exchange,  wc<?'ian?":i  t" 
multiply  the  number  of  francs  by  185,  :ii>d  divide  by  100  ;  rouoyr"'  '^'"" 
or,   what   is    equivalcntly  such  division,   we  point  ofl'   2 
decimals  and  have  the  answer  in  dollars  and  cents. 

16.  Change  25  francs  to  American  money,  at  the  par  of 
♦exchange. 

OPERATION-. 
25 

181- 

200 
25 
12J 


^1  [■  the  multiplication  by  J. 
Ans.  $4.68 1. 


15^  EXCIIANCr;. 

17.  Change  325  francs  to  Americim  money,  at  the  par 
of  exchange. 

18.  ('hange  560  francs  to  American  iuono3;,  at  the  ]>ur 
of  exchange. 

19.  What  will  be  the  value  of  a  bill  of  exchange  un 
Paris,  in  American  money,  for  8550  iVancs,  the  rate  of 
premium  being  5.12  francs  per  dollar?     yl»s.  $1(585.20. 

JV(/fij. — Hero,  the  exchangc=5fr.  12  centimes,  ha.'^  to 
be  addeil. 

20.  What  will  be  the  valu?'  of  a  bill  vi  exchange  on 
I'aris,  in  Aniorieao  money,  for  ,1500  francs,  there  being  a 
discount  on  French  exchange  of  5.13  per  dollar? 

A'utr. — Here,  the  exchange  has  to  be  subtracted. 

To  uhange  Kkmark. — When   it  is  required  to  change  American 

""ench"nu^ey.  i"'^°^'y  to  French,  the  process  is  reversed. 

21.  Change  S4.G82  to  French  money,  ut  the  par  of 
exchange. 

22.  C'haDge  $870.75  to  francs.  Ans.  4644. 

OPEKAilON. 

18i  870.75 
4  4 

75)3483.00(4644fr.  Ans. 
oOO 


4S3 
440 


330 
300  . 

300 

300 

Xotc. — For  the  convenience  of  the  division,  both  divitor 
and  dividend  are  changed  to  4thii.     This  enables  us  to 
avoid  the  fractional  f ,  but  does  not  change  the  value  of     " 
the  result. 

23.  Change  S75.0  to  francs,  adding  premium  5.15. 


GUAGINii.  loll 

24,  Wliat  will  be  the  value  of  a  bill  of  exchange  on 
Paris,  in  American  money,  for  8550  francs,  the  premium 
being  5.08  ? 


GUAGTNG. 

287.  Gv.<i(jing  is  the  process  used  to  find  the  contents  of  ouasmg  <ic- 
vessels  chiefly  of  curved  form.  '^"'■''^■ 

288.  Exactness,  because  of  the  different  curvatures  of Exft(tne.s,«  im 
staves,  in  finding  the  interior  measurement  of  such  vessels,  i"''^*i^'''- 

is  impossible ;  but  t!ie  approximation  to  it  is  sufficiently 
near  to  answer  all  commercial  purposes. 

289.,  The  mean  diameter  of  "a  cusk  is  found  by  adding  To  liud  iii.^ 
to  the  head  diameter  -j  of  the  difl'ercncc  between  the  bung  ""''** '^"""'''^'' 
and  head  diameter,  or  when  the  staves  are  not  greatly 
I'urved,  by  adding  -iV-     Tbis  gives  the  vessel  a  cylin- 
drical form.     V/e  then  multiply  the  area  of  the  base  by 
the  altitude. 

290.  To  find  the  solid  contents  in  cubic  inches,  wexofindth.-.soitH 
multiply  the  square  of  the  mean  diameter  of  the  cylin-  ^j'"j^';"'^  '"  '"" 
drical  vessel  by  the  decimal  ,7S54,  and  the  product  by  the 

length.     The  contents  in  gallons  are  found  by  dividing  by  t,^c  coni.Mits  i« 
231,  which  is  the  number  of  cubic  inches  that  expresses  saiions. 
a  gallon  of  liquid  measure.     (A'.    00.) 

291.  When  a  vessel  is  very  ir'-cgular,  or  when  a  cavity  ^vhon  a  VO-.W.I 
not  very  large  is  to  be  measui'  d,  the  easiest  and  mostj^  ^'^y  irroy.,- 
accurate  way  to  find  the  conteuts  is  to  fill  the  vessel  or 

cavity  with  water,  and  then  to  measure  this  put  in  casks 
whose  dimensions  are  found  as  above. 

Example  1. — How  many  gallons  are  in  a  cask  whose 
bung  diameter  is  40  inches,  head  diameter  28  inches,  and 
length  48  inches  ?  Ana.  211.50gal. 

KXPLAXATIOX. 

Having  found  the  differ-  Kxpiauation  oi 
'  ence   oi    the   diameter,    we  ' 

I  take  f  of  it  and  add  to  the 
h  head  diameter.  We  then 
1296x48x34=  2H.50gal.  I  multiply  the  square  of  the 
mean  diameter  (o6XoO),  the  length  (48),  and  34  together, 
and  point  off  4  decimal  places.  This  gives  the  answer  in 
gallons  and  decimals  of  a  crallon. 


OPERATION. 

40 

— 

28 

= 

12 

\ 

i 

12 

= 

1 

28^ 

+ 

8 

=:: 

36 

36 

X 

36 

= 

1296 

1<')<'  GUAGING. 

«<ni,iitsiii';."a-      Remark. — The  decimal  .7854  diviJed  by  231  (=the 
l?s'?>''(?i""""(i.li-  cubic  inches  in  a  gallon  liquid  measure),  carried  to  f(mr_ 
"■''  decimal  places,  is  .0034  ;  and  this  decimal  ^multiplied  by 

the  square  of  the  mean  diameter,  and   by  the  length  of 

cask,  gives  the  contents  in  gallons. 

2.  What  are  the  contents,  in  wine  gallons,  of  a  cask 
whose  length  is  .36  inches,  and  whose  head  and  bung 
diameters  are  each  IG  and  19  inches  '{ 


i\»    ti'  ni'.-n--- 

o    ;i    iMlI\<'i 


Note. — To  measure  curved  vessels,  simply  multiply  the 
length  by  the  square  of  the  mean  diameter,  then  by  34, 
and  mark  off  4  decimal  placfs. 

3.  What  arc  the  contents,  in  beer  measure,  of  a  cask 
whose  length  is  4G  inches,  and  whose  head  and  bung 
diameters  are  each  24  and  32  inches!'' 

KoU'. — Tn  bocr  measure,  the  decimal  .7854  is  divided 
by  282  cubic  iucht.s=:the  beer  gallon.     (Art.  94.) 

4.  What  are  the  contents,  in  bushels,  of  a  hhd.  whose 
length  is  48  inches,  and  wlin-c  bead  nnil  bmi"-  diniiftprs 
are  34  aud  42  inches  '{ 

.Voii. — In  dry  me;isurc,  the  bushel  equals  2150  cubM 
inches. 

5.  What  arc  the  contents,  in  bushels,  of  a  cask  whoe* 
length  is  44  inches,  and  whose  head  and  bung  diameters 
are  32  and  40  inches? 

().  What  are  the  contents,  in  wine  gallons,  of  a  cask 
whose  length  is  38  inches,  and  whose  head  and  bung 
diameters  are  each  15  and  IS  inches  ? 

7.  In  a  barrel  30  inches  deep,  and  its  diameter  (od« 
third  from  the  top)  20  inches,  how  many  wine  gallons? 
20x20=400.    400x30=12000 

34 


48000 
36000 


408000=408  lOgal. 
J\fote. — Multiply  the  diitmeter  (one  third  from  the  top) 


TONNAGE. 


161 


by  itself,  and  this  by  the  depth.     Then  multiply  by  34 
and  cast  oflF  4  figures  for  decimals, 

8.  In  a  barrel  34  inches  deep  and  its  diameter  18  inches, 
how  many  wine  gallons  ? 

9.  In  a  barrel  28  inches  deep  and  its  diameter  16  inches, 
how  many  wine  gallons  ? 

Note. — For  the  guaging  of  corn,  peas  and  potatoes,  see 
60-63  pages.  " 


TONNAGE. 


292.  The  Tonnage  of  a  vessel  is  her  measured  capacity  Tonnage  dc- 
of  freight.  fined. 

The  quantity  that  can  be  carried  is  estimated  by  two  Two  rules  t<> 
rules ;  one  known  as  the  carpenter's,  the  other  the  gov-  '"*''*  ^^^  *°°" 
ernment's  measures. 

Example  1. — What  \»  the  tonnage  of  a  single  decked 
vessel  whose  length  is  80  feet,  breadth  21  feet,  and  depth 
18  feet?  Aus.  318-,V  tons. 

OPERATION. 


nage. 


21 

80 

1680 

18 

13440 
1680 

95)30240(318-iV. 
285 

174 

95 

790 
760 

30 


12 


EXPLANATION. 

We  here  multiply  the 
length  of  keel,  breadth  at 
main  beam,  and  depth  of 
hold,  in  feet,  together,  and 
divide  by  95;  the  quotient 
is  the  number  of  tons. 


162 


TONNAG 


i 


To  measure  a       Note. — ^^For  a  double  decker  take   h  of  the  breadth  at 
double  decked  ^j^g  main  beaip,  for  the  depth  of  the  hold,  and  proceed  as 
above. 

2.  What  is  the  carpenter's  tonnage  of  a  single  decked 
vessel  whose  length  is  90  feet,  breadth  25  feet,  and  depth 
19  feet  ? 

3.  What  is  the  carpenter's  tonnage  for  a  double  decked 
vessel  whose  length  is  200  feet  and  breadth  38  feet? 

Am.  ]  520  tons, 

4.  What  is  the  gOTernmcnt's  tonnage  for  a  vessel  of 
single  deck  v.hose  length  of  keel  is  80  feet,  breadth  at 
main  beam  21  feet,  and  depth  18  feet? 

EXPLANATIOK. 

From  the  length  we  tak< 
f  of  tlu^  breadth  (SO— 12|), 
and  having  multiplied  the 
remainder  by  breadth  and 
depth,  divide  by  95  for  the 
result  required. 


Government 
measure  exam- 
ple. 

OPERATION. 

80 
12i 

67f 
21 

67 

134 

H 

14151 

18                              ! 

11320 
1415 
7i 


95)254  7  7i(268il^  tons. 
190     •  "' 


047 
570 


<  <  / 

7<.0 

17i 


flow  to  meais- 


.  Kemark. — By  government  rule,  for  a'  single  decker, 
take  length,  in  feet,  above  the  deck,  from  the  fore  part  of 
the  main  stem  to  the  after  part  of  the  stern  post;  the 


ANNUITIES.  16;) 

breadth,  at  the  widest  part  above  the  main  wales,  on  the 
outside ;  and  the  depth,  from  the  under  side  of  the  deck 
^lank  to  the  ceiling  in  the  hold. 

").  What  is  the  government  tonnage  for  a  vessel  of  the 


3  "capacity  as  the  one  given  in  3d  example  ? 


Remark, — To  measure  a  double  decked  vessel,  take  the  to  medsure  a 
length  above  the  upper  deck  ;  for  the  depth,  take  \  the  do"W«  d^ci^^'- 
width,  and  proceed  as  before  directed. 

6.  What  is  the  carpenter's  tonnage  for  a  single  decker 
whose  length  is  100  feet,  breadth  25  feet,  and  depth  20 
feet  ? 

7.  What  is  the  government's  tonnage  for  the  same 
r:i;.;icitv?  Ans.  447-iV  tons. 


ANNUITIJ]S.  r 

293,  An  Amiuity  is  a  sum  of  money  payable  to  a  person  Annuity  «^- 
for  a  certain  term  of  years  ;  usually,  for  the  life  time.  ^^-      \ 

294,  An  annuity  not  paid  at  the  stipulated  date  is  said  Aif  annuity  in 
to  be  iu  arrears  ;  when  it  is  not  to  commence  until  some  'i"p'»''s; 
future  time,  it  is  called  a  reversionary  annuity;  but  when  j^^^  reversion; 
its  payments  have  commenced,  it  is  said  to  be  in  posses- j^^      g^g^j^^jj 
sion. 

295,  Annuities   are   often  bought   and   sold,  as  other  Bought  arid 

•    1        1  gold. 

commercial  values.  ■  j 

296,  The  sum  of  annuities,  such  as  rents,  pensions,  tj,^,  amount  of 
salaries,  remaining  unpaid,  with  the  interest  on  each,  is  "■'^  annuity, 
called  the  amount  of  the  annuity.  \% 

297,  To  find  the  value  of  an  annuity  iu  arrears  observe  ,p^  ^^^^  j,,^ 
simply  the  method  to  ascertain  an  amount  at  interest ;  but,  talue. 

for  an  expeditious  way  to  discover  the  same,  we  give  the 
followiusc : 


i«;4 


ANNUITIES. 


Tal'lo    shovriufi 
ciertain 


TABLE, 

29$.  Showing  the  amount  of  the  annuity  of  $1,  £1,  etc.,  at 
4,  5,  0  and  7  per  cent,  compound  interest,  for  any  number  of 
years  not  exceeding  20. 


Years 

4  per  cent. 

5  per  cent. 

6  per  cent. 

7  per  cent. 

1 

1.0000 

1.0000 

1.0000 

1.0000 

2 

2.0400 

2.0500 

2.0600 

2.0700 

3 

3.121G 

3.1525 

3.1836 

3.2149 

4 

4.2464 

4.3101 

4.3746 

4.4399 

5 

5.4163 

5.5256 

5.6370 

5.7507 

6 

6.6329 

6.8019 

6.9753 

7.1532 

7 

7.8982 

8.1420 

8.3938 

8.6540 

8 

9.2142 

9.5491 

9.8974 

10.2598 

9 

10.5S27 

11.0265 

11.4913 

11.9779 

10 

12.0061 

12.5778 

13.1807 

13.8164 

11 

13.4863 

14.2067 

14.9716 

15.783S 

12 

15.0258 

15.9171 

16.8699 

17.8884 

13 

16.6268 

17.7129 

18.8821 

20.1406 

14 

18.2919 

19.5986 

21.0150 

22.5504 

15 

20.0235 

21.5785 

23.2759 

25.129Q 

27.888(P 

16 

21.8245 

23.6574 

25.6725 

17 

23.6975 

25.8403 

28.2128 

30.8402 

18 

25.6454 

28.1323 

30.9056 

33.9990 

19 

27.6712 

30.5390 

33.7599 

37.3789 

20 

29.7780 

33.0659 

36.7855 

40.9954 

E.iplonation  of 
method. 


Example  1. — "What  is  the  amount  of  an  annual 
pension  of  S500  in  arrears  for  16  years,  at  7  per  cent,  com- 
pound interest  ?  Am.  813,940 

Note. — The  table  shows  the  7  per  centage  of  16  yeari 
to  be  27.8880 ;  this  multipled  by  the  amount,  500,  gives 
139440000=813,940. 

2.  What  is  the  amount  of  an  annual  salary  of  81500, 
which  has  been  in  arrears  for  10  years,  at  7  per  cent.  ? 

3.  What  is  the  present  worth  of  an  annual  pension  of 
$120,  at  6  per  cent.,  to  continue  3  years  ? 

Am.  $320.76. 

Explanation.— We  find  by  the  table  that  8120= 
8382.03;  this  divided  by  the  amount  of  ?1,  compound 
interest,  gives  the  quotient  of  present  worth.  This  in 
evident,  since  the  quotient,*  multiplied  by  the  amount  ot 


ALLiaATTOX.        '  ll^f 

^1  for  3  years,  at  compound  interest,  is  $382.03.  By 
reference  to  the  tabic  under  the  article,  Interest,  the 
amount  of  ^1  at  compound  interest,  is  seen  to  be  $1.1910; 
then  3382.03  divided  by  it  gives  S320.76. 

4.  "What  is  the  present  value  of  an  annuity  of  $300,  at 
7  per  cent.,  to  continue  10  years? 

5.  What  is  the  value  of  a  pension  of  890  per  annum, 
la  arrears,  for  12  years,  at  7  per  cent.  ? 

6..  What  is  the  value  of  a  ground  rent  of  S200  per 
:innum,  for  8  years,  at  7  per  cent.  ? 

»     7.  For  what  can  I  buy  with  ca.sh,  an  annuity  of  $300, 
to  continue  5  years  ? 

8.  If  you  lay  up  $50  a  year  from  the  age  of  21  to  that 
of  60,  what  will  be  the  amount  at  compound  interest  ? 


ALLIGATION. 

i99.  AlU<jat)(in  is  a  mercantile  usage  to  procure,  by  the  All  gation  d*- 
mixture  of  simple  substances  of  difterent  qualities,  a  com-**      " 
pound  of  intermediate  value. 

300.  When  the  quantities  and  prices  are  known,  the  When  called  a;  ■ 
employed  process  is  coHXq^  Alligation  Medial;  but  when  '^* '"^ 
the  proportional  quantities  of  the  mixtures  are  not  known,  ^^^^  ^u^^ 
except  through  their  mean  rates,  it  is  called  Alligationnake. 
.ilfernate. 

Example  i. — A  wine  merchant  mixed  together  3 
difterent  kinds  of  wine,  as  follows  :  3  casks  at  $40  per 
caakj  3  at  $50;  and  3  at  $60:  what  is  the  mean  price  per 
cask?  Ans.  $50. 

EXPLANATION. 

Having  multiplied  the  cost  To  find  tiio. 
of  each  cask  by  3,  and  found  ^*''^'^- 
by  addition,  the  united  worth, 
we  say,  as  the  sum  of  all  the 
quantities  (9)  is  to  their  whole 


OPERATION. 

•40      50      60 
3        3        3 


120+1504-180=$450 

then  9:450::1:50 
value  (450),  so   is   any  part    (1)    of  them  to  its   mean 
price. 

2.  A  grocer  mixed   together  three  different  kinds  of 
coffee,  as  follows :    1001b.  of  Rio,  at    10  cents  per  lb. ; 


]Gt) 


ALLIGATION   ALTERNATE. 


2501b.  of  Mocha,  at  20  cents  per  lb. ;  and  3001b.  of  St. 
Domingo,  at  8  cents  per  lb. :  what  is  the  mean  pryie  of 
the  mixture  't 

3.  A  grocer  mixed  together  three  different  kinds  of 
sugar,  as  follows  :  45Ulb.  at  G  cents  per  lb. ;  740  at  7  cents 
per  lb. ;  and  5001b.  at  8  cents  per  lb. :  what  is  the  value 
of  one  pound  of  the  mixture? 

4.  A  goldsmith  mixed  three  different  kinds  of  gold,  in 
these  proportions  :  4oz.  of  14  carats  fine ;  5oz.  10  carats ; 
and  6oz.  18  carats :  how  many  carats  of  fine  gold  are  in 
one  ounce  of  the  mixture  ? 

5.  A  grocer  mixed  40  gallons  of  water  with  150 
gallons  of  wine,  valued  at  ^1  per  gallon:  what  is  the 
value  of  the  mixture  y 

0.  A  grocer  mixes  together  two  different  grades  of 
mola.sses  :  the  first  is  worth  28  cents  per  gallon,  but  the 
second  35.  The  mixture  contains  Igal.  of  the  first  kind  to 
2gal.  of  the  second  :  what  is  the  mixture  worth  per  gallon  '( 

7.  What  wonld  the  following  mixture  of  corn  be  worth  a 
bushel :  20  bushels  at  78  cents  per  bush. ;  25  at  65  cents 
per  bush. ;  and  15  at  45  cents  per  bush.  ? 


ALLIGATION    ALTERNATE. 


^UipitionAittr-      301.  AUiijntion   Alttrnate   is  a  process   showing   what 
n«f«  detiiiod.     quantity  or  quantities  of  known  values  must  be  mixed  with 

a  given  quantity  of  another,  to  produce  a  mixture  of  a 

specified  name. 

Example  1. — A  grocer  wishes  to  mix  wines,  which  ar<> 

14,  IG,  22  and  28  shillings  a  gallon,  in  such  form  that  the 

mixture  shall  be  worth  20  shillin.2;s  a  gallon. 

OPERATION. 


How  tofiod  t)ie 
worth  of  mix- 

nirog. 


8 

2 

4 

128 

6 

8  gals,  at  148 
2     "     ♦'  16s 

4     "     "  22s 

6     "     " 

28s 

Ans. 


EXPLANATION. 

Here  we  write  the  prices 
of  the  different  wines  in 
order,  in  a  vertical  form; 
and  each  price  that  is  lees 
tbau  that  of  the  mixture  we 
connect  by  a  line  with  one 
or  more  of  the  prices  that 
are  greater  than  it.  We 
then  write  the  difference 
between  the  required  price 


TARE  OR   ALLOWANCE.  1  67 

and  each  of  those  that  arc  less,  by  the  side  of  the  larg'er 
one,  with  which  it  is  connected;  also,  the  difference 
between  each  larger  price  and  the  price  of  the  mixture  by 
the  side  of  the  less,  in  its  connection.  The  difference 
standing  by  the  side  of  each  price  is  the  quantitj'  requi^od 
to  form  the  mixture.  * 

Note. — By  calculating  the  mean  prices  as  given  above=  p,.ooi 
400,  and  dividing  by  the  g4llons=20,  we  find  by  the  quo- 
tient 20  tlio  proof  of  the  correctness  of  the  calculation. 

2.  A  grocer  wishes  to  mix  4  different  kinds  of  wine,  to 
obtain  a  mixture  that  shall  be  worth  81  per  gallon.  The 
first  kind  is  worth  75  cents  per  gal. ;  the  second,  50  cents  ; 
the  third,  81.25;  and  the  fourth  §2:  how  many  ;r:illnns 
must  he  take  of  each  kind  ? 

•J.  A  grocer  wishes  to  mix  four  different  sorts  of  tea, 
worth  5s.,  6s.,  8s.  and  9s.  per  lb.,  so  as  to  have  a  mixture 
of  871bs.  worth  7s.  per  lb.:  how  much  must  he  take  of 
each  kind  ?  Am.  21!,' lb. 

4.  A  goldsmith  has  4  different  kinds  of  gold,  14,  16.  20 
and  22  carats  fine  ;  he  wishes  to  obtain  from  them  a  mixture 
1 8  carats  fine  :  how  many  ounces  must  he  take  of  each  't 

.5.  A  drover  has  sheep  worth  $2,  S4,  86,  88  and  $10 
each :  how  m my  from  these  must  he  take  to  form  a  flpck 
of  SO  wortli  $11  each? 

6.  A  grocer  wishes  to  mix  3  different  kinds  of  su;.;ar,  to 
obtain  a  mixture  worth  8  cents  per  lb.  The  first  kind  is 
worth  7  cents  per  lb.;  the  second,  6  cents  ;  and  the  third, 
10  cents:  how  many  of  each  must  he  take  to  form  the 
mixture  ? 

7.  A  grocei*  wishes  to  mix  4  different  kinds  of  coffee,  so 
as  to  have  a  mixture  worth  12  cents  per  lb.  The  first  is 
worth  8  cents  per  lb.;  the  second,  10  cents ;  the  third  and 
fourth  14  cents  :  what  quantity  must  he  take  of  eacli  ? 


TARE  OR  ALLOWANCE. 

JJOiS.    fare  is  a  mercantile  term  to  express  an  allowance  ,^^^^  ,i*h„.,,<. 
made  in   the  transfer  of  goods  for  known  or  presumed 
•deficiencies. 


168  TARE  OR  axlowancj: 

An  allowance  .it      303.  An  allowance  is  made  at  tlie  port  of  entry  on 
ports  of  entry,  (juti^ljle  goods,  because  boxes,  etc.,  containing  imported 
articles,  are  not  considered  any  part  of  the  commodities 
subject  to  tariff  charge. 
Allowance  by        304.  Ecsidcs   this   allowance  or  tare,  Reductions  arc 
an  e.       ^^de  by  wholesale  merchants  on  boxes,  etc.,  in  the  inter- 
changes of  trade. 
The  allowance       305.  When  goods  are  not  actually  entitled  to  theb( 
of  draft.  allowances,  a  deduction,  comnjercially  known  as  draft,  h 

made  for  waste.    This  is  9  lbs.  on  the  ton,  and  proportion- 
ally for  sniaaller  weight. j 
Allowance  for       30(».  On  liquor  in  casks  there  is  usually  the  allowance 
ikjuor*.  pf  5  pgj.  ^■Qnt.  for  leakage.     On  liquor  in  bottles  (particu- 

larly.  porter  and  other  fermenting  liquors),  for  breakage", 
there  is  an  allowance  of  10  per  cent. 
fcrofK  weight;       307.  (jross  weight  is  the  whole  weight  of  the  good5 
Ni-t  woicht       before  allowance  is  made  ;  and  net  weight  is  what  remain.* 
after  allowances  are  deducted. 

Example  1. — What  is  the  net  weight  of  10  boxes  of 
candles,  each  weighing  501b8.,  there  being  a  tare  of  41bh. 
on  each  box  ?    ' 

OPERATION. 

50  10 

10  4 


.  Note. — Examples  in  Tart- 
are  performed  by  multipli- 
cation, subtraction,  and  in 
some  cases,  by  proportion. 


r)001bp.      401bs.  tare. 
40 

460  net. 

2.  At  §156  per  hhd.,  what  will  4hhds.  of  sugar  amount 
to,  allowing  lOlbs.  per  cwt.  tare  on  the  gross  weight  of 
19cwt.,  3qrs.  151bs.  ? 

3.  At  16  cents  per  lb.,  what  is  the  worth  of  4  bags  of 
coflFee,  the  gross  Veigbt  of  which  is  6501b8.,  allowing  2 
lbs.  tare  on  lOOlbs.  ? 

4.  At  8  cents  per  lb.,  what  is  the  cost  of  Shhds.  of  sugar, 
weighing  gross,  Icwt.,  3qrs.  14lbs.;  2ewt.,  2qr8.  131bs.  ; 
3cwt,,  Iqr.  151bs.,  allowing  tare  of  91bs.  in  each  cwt.? 

5.  .At  §15  per  cwt.,  what  will  lOcwt.,  3qrs.  121bs.  sugar 
cost,  allowing  tare  lOlbs.  per  cwt.  ? 

ATo/e. — Deduct  tare  from  gross  weight,  and  find  answer 
by  proportion  :  if  lOOlbs.  cost  $15,  what  will  the  number 
of  lbs.  net  weight  cost  ? 


SINGLE   rOSlTION.  169 


SINGLE  rpSTTION. 

308.  Single  Position  is  a  jurocess  to  ascertain  the  true  Single  Position 
answer  to  a  question,  by  assinnuiL^a  certain  number  as  the   *  °® 
rightful  one,  or  as  Icadin-i  tn  i:. 

ExA.\iiB.E  1. — I  liave  a  c<'ifain  number  of  sheep  in  my 
pasture,  and  if  the  number  weio  increased  by  J  and  J,  the 
whole  would  be  06  :  what  i.s  the  number  '(  Ans.  36. 

OI'KI.'A  I  |(l.\. 

Suppose  12  is  the  number,  then  by  adding  \  of  12=6, 
and  i  ot  1.2=4  to  12=22,  wo  find  the  supposed  number 
was  the  true  one  by  the  proportion,  22  :  66  ::  12  :  36. 

Remark. — The  operation  with  1 2,  which  gives  as  ab6te, 
22,  enablps  us  to  obtain  the  evidently  true  answer  in  the 
fourth  term  of  the  proportion. 

309.  The  examples  in  siii;^le  position  can  be  analyzed 
without  trouble.  Thus,  in  the  above  c.\;imple,  the  number 
to  be  found  is  fractionally  ^=1  :  i  of  f=^  and  i=f ;  which 
added=-^^'-=UG.  Hence,  ii  -';.'•-:=()(),  -,\=6,  or  -^  of  the 
number;  then,  asf=the  whole,  iix6=o6.  Ans. 

2.  The  master  of  a  school  being  a.sked  how  many  pupils 
he  had  charge  of,  said  tliiit  if  he  had  as  many  more  as  his 
present  nuaiyer,  \  as  many  more,  J  and  \  as  many  more, 
he  should  have  296:  what  wis  the  nuuiber  '(      An^.  i>6. 

3.  A  man.  who  was  asked  his  ago,  said\  that  if  -f  of  it 
were  muitiplied  by  7,  and  -g-  subtracted  from  the  product, 
the  remainder  would  be  66  :   what  was  his  age  ? 

4.  What  is' the  number,  which  multiplied  by  7,  and  the 
product  divided  by  6,  will  have  a  (juotient  of  14  't   Ans.  \  2. 

5.  Two  men  have  the  same  income.  One  saves  i  of  his, 
but  the  other,  by  spending  twice  as  much,  finds  himself,  at 
the  end  of  4  years,  $560  in  debt:  what  is  the  annual 
income  ?  Ans.  $420. 

6.  Throe  speculators  gained  $2400,  of  which  B  took  3 
times  as  much  as  A,  and  C  4  times  as  much  as  B:  what 
was  the  (share  of  eachf 

13 


170  DOUBLE   POSITION. 


DOUBLE  POSITION. 

€> 
Double  Posi-        310.  Dov.bJe  Position  is  a  method  to  determine  an 
tion  defined,     a^ng^yej.  \)y  the  use  of  two  munbers,  supposed  to  .bo 

the  ones  sought.  • 

Results  differ        311.   In  this,  the  results  vary  fj'om  Single  Position 
sim"ie^PMi-^^^y  ^^'^^  beilig  proportional  to  the  assumed  numbers. 
won.  Example  1. — A  person  having  a  certain  sum,  spent 

$100  more  than  -J-  of  ityand  had  reiiiaining  $40  more 

than  ^  of  it :  wliat  had  he  at  first  ? 

OPEUAflON. 

Suppose,  first,  he    had     §150.0,   then 

§100   more   than  •^=    •400=  sum  spent, 


and  1100=remainder.; 

hut  $40  more  than  ^=    795  ' 

■  hence,  |305=lst  error. 

Suppose,  second,  he  h^d    $2000,  then 

$100   more   than   ^=    500=sum  spent, 

and  1500=remainder ; 

but  $40  more  than  1=- 1040 

hence,  $4,60=  2d  error. 

$1500x4GO:=S690000=rlst  assumed  No.x2d  error. 
$2000x306— $(310GuO=:2d  assumed  Ko.Xlst  error. 


155  )  $80000(|^51G-i^V-  ^n^- 
lib 

\    250    . 
^     155 

-      950  ', 

930 

•  20 
that  is,  the  difl'crence  of  the  products  divided  by  the 
difference  of  the  errors,  ^ives  the  result  of  $51()-iVs-- 


DOUBLE   POSITION  171 


KXPLANATION. 

By  the  cxiimplo  it  will  be  noticed,  that  1st,  having  Process  oi 
yiipposed  two  nnnibers,  wc  proceed  with  them  accord-^'''"""'*' 
jng  to  the  question  ;  2d,' that  having  compared  each 
result  with  that  in-  the  question,  and  calling  each  dif- 
ference an  error,  we  nuilli])lied  the  1st  :)ssunied  "^i^n- jlj^exarnpu-" 
ber  by  the  2d  error,  and  the  2d  assumed  number  by 
the  first  error;  andJkl,  that  wc  divided  the  diiference 
of  the  products  by  the  dino'-"'>  ■■■  of  ■(li'^  r>i-ror..  fny  t],.-. 
true  answer. 

Remark — Had  one  assumed  numbq^  been  too  great  whon  iho  w 
and  the  otficr  too  small,  we  should  have  divided  the  be?^*^are' t™ 
Piim  of  the  products  by  the  sum  of  the  errors.  |_«^ff^i  ^^  '«'* 

■J.  Three  perilous  speaking  of  their  ages,  B  said.that' 
was  10  years  older  than  A,  and  0,  that  his  doubled 
both  of  theirs:  what  were  their  respective  ages*  the 
united  sum  being  lUO  ?       Afi^.  A's  20,  B's  30,  C's  50. 

o.  What  number  is  that,  which  being  divided  by  7, 
.'I'ld  the  quotient  (liininislu  .1  by  10,  throe  times  tlic 

iuainder  is  2-4i'  '    J.ns.  126. 

i.  Two  clerks  haw  .;..  .j^mo  income;  A  saves  }  of 
Ills  yearly,  but  J5,  by  improvidently  spending  5>150 
per  annum  more  than  A,  at  the  end  of  8  years  finds 
himself  8-400  in  d^bt :  what  are  their  incomes,  and 

hut  the  annual  expenditures  of  each  ? 

Note. — First,  assume  that  each  had  S200;  second,'. 
-iiO  ;  then  the  ei-n>rs  will  bo  400  and  200.  • 

J/N.  e;  .  sexp-.^00;..B's§450. 

5.  A  ])lanter  purchased  a  number  of  hcraes,  mules 
and  cows  for  $ii340.  lie  paid  for  each  horse  $50;  tor 
>  ch  mule  f  hh  much  as  for  a  horse;  and  for  each 
^o\v  ^  of  tlie  price  of  a  horse.  There  wore  o  times  as 
many  mules  as,  horsps,  and  twice  as  many  cows  as 
mules  :  Vs-hut  was  t';  '••  ••^- 'I'^r  ot*  each  ? 


172  MISCELLANEOUS    EXAMPLES, 


MISCELLANEOUS  EXAMPLES. 

312.  Example  1. — A  owes  B  $iOO,  tlue  in  3mo., 
$250  in  Inio.:  what  is  the  mean  time  of  paj'ment? 

2.  What  is  the  bank  discount  of  $455.GLt,  payable  in 
6mo.,  at  the  nato  of  7  per  cent.  ? 

3.  What  commission  is  a  factor  to  receive  on  the 
sales  of  35ijhh(l.  of  sugar,  at  ^62  per  hhd.,at  the  rate 
of  2^  per  cent.,  and  2|-  per  cent,  for  the  guarantee 
of  sale  ?  • 

4.  A,  B  and  C  jointly  purchased  a  piece  of  land. 
A  paid  I  of  the  price,  B  i,  and  C  the  remaining  i. 
They  subsequently  determined  to  dispose  of  it,  and 
gained  $7riO  :  what  was  the  gain  of  each  ? 

5.  A  grocer  wishes  to  mix  three  grades  of  sugar 
to  obtain  a  n\ixture  wo»th  8  cents  per  lb.  The  first 
is  worth  6,  the  second  7,  and  the  third  10  cents: 
what  must  he  take  of  each  to  obtain  the  desired 
kind  ? 

6.  What  is  the  amount  of  $375,  at  7  per  cent, 
interest,  for  lyr.  Gm.?" 

7.  Wlut't  is  the  amount  of  |!400,  at  7  per  cent,  com- 
pound interest,  for  4  years  ? 

8.  A  trader  purchased  400  barrels  flour,  at  $4.50 
per  bbl.  payable  in  Gm.  In  order  to  gain  10  per 
cent,  and  give  a  credit  of  8m.,  reckoning  bank  dis- 
count at  7  per  cent.  j)er  an.,  what  must  he  ask  per 
bbl.? 

9.  There  is  a  fish  whose  head  weighs  141bs.,  his 
tail  weighs  as  much  as  his  head  a«d  -i\  as  much  as 
his  body,  and  his  body  weighs  as  much  as  his  head 
and  tail :  what  is  its  weight?  Ans.  801bs. 

10.  A,  B,  and  C  form  a  partnership.  A  puts  in 
$500  for'  3m.,  B  S600  for  2|-m.,  an<l  C  $300  for  5m. 
They  gain  $550 :  what  is  the  share  of  each  ? 

11.  A  nicroJaant  ovves  $G00,  due  in  7m. ;  but,  as  he 
pays  ^  cash,  and  will  pay  i  in  4m.,-iiow  long  .may  he 
retain  the  balance  ?  Ans.  2yr.  lOra; 


iMISCELLANEOUS    EXAMPLES.  ITo 

12.  A  person  buys  a  farm  for  $2000,  paj-ablc  in  16 
.  equal  annual  instalments,  cojnmencing  thefirst  year 

after  the  purcliasc.     At  the  ra{c  of  6  per  cent.,  what 
sum' paid  down  would  fulfil  his  engagement? 

A>.s.  S1263.23. 

13.  At  12  per  cent,  advance,  what  will  S750  be  in 
£  s.  d.? 

14.  A  watch  sold  for  $60  lost  12  per  cent,  by  the 
'  sale  :  what  was  the  cost? 

15.  At  £'.)o  per  cwt.,  what  is  the  value  of  611bs.  ? 

16.  Wiiat  is  the  interest  of  $2650,  at  7  per  cent.,  for 
90  days  ?  • 

17.  What  is  the  government  tonnage  of  a  single 
decker  whose  length  is  50  feet,  breadth  12}  feet  and 
depth  10  feet? 

18.  What  are  the  contents,  in  gallons,  of  a  cask 
(  whose  length  is  44  inches,  head  diameter  28  inches, 

and  bung  diameter  86  inches? 

19.  A  owes  B  $800,  payable  July  4,  1863,  and  B 
■  owes  A  $60(>,  payable  October  6,  1863  :  when  is  the 

equated  time  for  settlemQut,  and  what  should  A  pay 
at  that  date  ? 


RADICAL    ARITHMETIC, 


PART    SIXTH. 

INVOLUTION 


inToitttion  de-  gl3,  Tnvolution  is  the  process  iised  to  raise  a.  iiura- 
*"°  ber  to  some  required  power;  tlui's,  2x2=4,  the  2d 

power  or  square  of  2  5  2x2x2=8,  the  3d  power  or 

cube  of  2. 
What  the  pow-     314.  The  power  is  the  product  of  a  number  multi- 
'^^^  plied  by  itself; 

Tke  intiex  or  315.  Sometimes  the  power  is  indieutcd  by  means 
*sponeat.         ^^  ^  sraall  figure  placed  at  the  right  hand  of  the  root, 

slightly  elevated,  called  the  index  or  exponent;  thus, 

3x3  is  written  3°,  and  is  read  the.  square  oi-2d  power 

of  3.  ' 

The  root,  or  bar  BIS.  Tlio  tsumber  at  the  basis  of  the  operation  *is 
isl  powerT'  known'as  the  root  or  the  1st  power;  when  once  mul- 
equarc;  '  tiplied,  it  is  known  as  the  2d  pow^cr  Or .  Square  ;  when 
cnbe.  twice  multipfiod,  as  the  3d  power  or  cube,  and  so  oh. 

What  the  ^  3li7.  The  squarp  of  a  number  consLsts-  of  twice  as 

cube^coasist  of.  JT^s'^y  figui'GS  as  the  root,  or  of  one  less  than  twice  as 

many;  the  cube  of  a  number  of  three  times  as  many 

as  the  root,  or  of  one  or  two  less  ihan  three  times  as 

many. 
...      ,  _  S18.   A   short   horizontal  line  over  two  or  more 

VmCUlUin;  ^  .  ^^        ,  •  1  IT-  ,1  • 

figures  IS  called  a  vinculum;  and,  like  a  parenthesis, 
Paren.aesis.     jiKjieates  that  the  numbers  so  connected  are  subject 

to  a  similar  operation;  thus,  8+2x8.  is  the  same  as 

(8+2^)x3=30.  ,     .    ' 


INVOLUTION.  175 

8i9t  A  numbt^r  already  raised. to  a  power  is  involved  The  index  mui- 
by  multiplyine;  its  index  by  the  index  of  the  po.wcrJiP^.'^'^^y*''^"- 
to  wliich  i-t  is  to  be  raised.- 

320.- A  vnl<>:ar'fraetion  is  involved  by  involving»the  How  a  vulgar 
nlmierator  and  denominator  separately;  thus,  the  SdyoWed. 

power  of  i  i8<ij3=(|)8=|^=|. 

321.  The  number  of  decimal  places  in  the  power  What  the  nun, 
of  a  deoimM  fraction  equals  the  number  of  decimarpj^ccs  ^qua™!* 
places  in  the  root  multiplied  by  the  index  of  the 

power;  thus,  the  od  power  of  .12  will  contain  0 
decimal  places,  for  .12x.l2=.0144xl2=.001728',  or 
123=.001728.   •  ^^ 

322.  To  divide  a  power  of  any  number  by  any  to  divide  pow 
other  ])ower  of  the  same  number,   we  subtract  the^'^. 

index  of  the  divisor  from  that  of  the  dividend;  thus, 
57-T-53=r,^ 

BxAMPLF.  1, — What  is  the  3d  power  of  4  ? 

Ans.  64. 

OPKRATION. 

4x4X4=64,  or  4x4=16x4=64. 

2.  What  is  the  6th  power  of  5  ? 

3.  Wliat  is  the  4th  power,  of  3  ?  Ans.  81. 
3x8x8x3=81,  or  8x3=9x3=27x3=81. 

4.  What  is  the  square  of  12  ? 

5.  llow  much  is  the  square  of  10  ? 

6.  llow  much  is  10  square  ? 

7.  How  much  is  lO'^  ? 

8.  AVhat  is  the  product  of  6^X6*  f 

9.  What  is  the  third  power  of  G*'  ?       A)is.  -gf^. 

10.  \\Miat  is  the  square  of  n 

11.  What  is  the  R(piare  of  5.^?'  Ans.  30}, 

.  5>  =  xxxV-=-^i-L=30l. 

12.  What  is  the  value  of.  lU*  ? 

18.  \Vhat  is  tlje  cube  of  3  ?  Ans.  27. 

'        •      '  ■  3x3x3=27. 
14.   What  is. the  cube  of  4?  « 

l.i.  What  is  the  cube  of  25  ?  ♦ 

16.  What  is  the  S(piare  of  IG-J  ? 

17.  Wiiat  is  the  square  of  .25  ?  Ans.  .0625. 

18.  How  many  figures  are  in  the  cube  of  99  ? 

Aiis.  6. 

19.  How  many  figures  arc  in  the  cube  of  243  ? 

20.  How  many  figures  arc  in  the  5th  power  of  99  ? 


176 


EVOLUTION. 


323. 


A   TABLE   OF   POWEKS. 


1st,  1 

2,    3 

4 

5 

6 

7i       8 

» 

2d, 

4 

9 

16 

25 

36 

49       64 

81 

3d, 

8 

27 

64 

125 

216 

343      512 

,  729 

4th, 

IG 

81 

256 

626 

1296 

2401  ,    4096 

6561 

5th, 

32 

243 

1024 

3125 

7776 

16807     32768 

59049 

6th, 

64 

729 

4096 

'  15625 

46656 

117649    262144 

.  531441 

7  th, 

J 

128 

2187 

16384 

78125 

279936 

823543   2097152 

4782969 

8th, 

266 

6561 

65536 

390625 

1679616 

5764S01   16777216 

43046721 

8th, 

.')12 

19683 

262144 

1953125 

10077696 

40353607  1342177*^8 

38742048* 

10th, 

1024 

69049 

1048676 

9765625 '60466176 

282475249  1073741824,3486784401 

EVOLUTION. 


Evolution  de- 
fined. 


iadicated. 


The  index  of 
the  root. 


324.  Evolution. — a  process  the  inverse  of  Involu- 
tion—is used  to  find  the'root  from  t]ie  giveo  power, 
and  may  be  defined  the  Extraction  of^oots. 
Howtherootis  325.  The-root  is  indicated  by  the  employment  of 
what  is  called  the  fractional  index,  or  by  this  symbol, 
which  is  known  as  the  radical  sign. 

326.  A  figure  placed  above  this  sign  is  the  index 

of  the  root,  and  is  the  same  as  the  denominator  of 

the  fractional  index."    When  a  number  is  not  given 

in  connection  with  the  radical  sign,  2  is  to  be  under- 

stood. 

A  power  and  a     3'^'S'-  A  power  and  a  root  can  be  indicated  at  the 

root  indicated  gj^j-^e   time   by  the   index   and  radical   sign ;    thus, 

together.  o^g5^32^  and  is  the  cube  root  of  the  5th  power 

of  8.  4       a      u 

im  effect  '^'^^'  Some  numbers  cannot  be  extracted,     feuch 

pTvversf  are  called  imperfect  powers,  and  their  roots  irrational, 

surd  numbers,  radical  or  sur  i  numbers.     For  convenience,,the  term 

radical  is  employed,  as  before  stated. 
Perfect  powers;     329.  Those  are  known' as  perfect  powers  that  can 
tutionai  roots.  %q  extracted,  and  their  roots  are  called  rational. 
The  first  ten        33©.  The  first  ten  numbers  and  their  squares  are, 

numbers   and  j    2,  3,     4,      5,     6,      7,      8,     9,     10, 

the.  squares.  ^,  ^,  ^.  ^^>^  ^J^  ^^^  ^^^  ^^^  g^^  ^^^ 

explanation  of     331.   The  numbers  in  the  first  line  are  the  square 
aboyenurai.ers.  j,QQt8  of  those  in  the  second;   the  numbers  m  the 
second  line  are  called  perfect  squares. 


EVOLUTION. 


177 


u:'!uRAT10N. 

10  2i(32 

9 

62)1  24 
12i 


Example  1 . — What  is  the  Bouarcroot  of  1024?  Ans.B2.  Process  ox- 

^  plained. 

EX  I"  L.\  NATION.  ^ 

Wc  first  point  off  tho  - 
number  into  periods  of 
two  figures  each,  as  We 
wish  to  find  the  squares 
of  the  tens  andhundreds. 
We  then  find  the  i^rcatest 
square  in  the  1  ,  ■\vliicli=3  tens  or  30.  Squiiring  tho 
3,  which  gives  *.i  .hundred,  wo  place  the  9  beneath  the 
hundreds  aijd, subtract;  this  takes  away  the  square 
of  the  tens  and  leaves  124.  ,  Next,  wo  double  tho 
divisor,  which  js  the  root  already  found,  and  divide 
this  remainder — without  tho  right  hand  figure — by 
it,  and  have  in  the  quotient,  the  uniks-figure  of  the 
root.  Annexing  this  figure  to  the  increased  divisor, 
we  multiply  again,  and  find  thudesiretl  result. 

Note.— A  similar  course  is  to  be  pursued  should 
there  be  more  iii-ures. 


ex- 

prri.ssiou  of  evo- 


To  extract  the  square  root  of  a  number  is  simply  a  simple  < 
to  resulve  it  into  two  equal  factors  ;  that  is,  to  find  aiution""" 
number  Aviiich,  multiplied  into  itself,  will  produce 
the  "iven  number. 

2.  How  large  a  square  floor  can  be  laid  with  576 
square  fdet  of  boards? 

OPKKATION.  EXPLANATION 

576(^24  Here,   the   area   beiu 

4  given,  we  are  to  find  the 

length  of  one  side. 


41)176 
176 


Remark  . 


Fie.  1 


To  make  the  operation  intelligible,  we 
form  a  square  whose  sides  shall 
be  2  ten8=20  feet  in  length. 
T4ie  area  of  this  square,  20x20 
=400  square  feet,  and  this 
number  taken  from  576  leaves 
176  to  be  employed  in  the  re- 
quired enlargement.  This  is 
made  on  the  2  sides,  as  in  fig- 
ure 2d  (though  it  could  be 
made  on  the  four),  and  their 
breadth  is  alike,  oeing  twice 


178 


^'\ 


\^^ 


EVOLUTION. 


When  the  pro- 
duet  of  divisor 
is  in  excess. 


4  ft. 


the  tens^f  the/oot=4  or  40  feet.  Now,  176  divided 
by  4i.i^'ites  4  feet,  Avhich,' added  to  the  trial  diyisoE, 
4(J==4f,  and  it?  the   entire  length   of  the  two 'sides, 

and  44x4=176;  that,  is, 
the  length  of  the  addi- 
tion multiplied  by  ite 
breadth  gives  its  area. 
Squaring  the  sides,  eacl. 
being  ^4,  shows  576,  and 
proves  the  sum. 

Tfhen  tlie  product  of 
the  divisor  by  a  number 
is  in  excess  of  the  divi- 
dend, mak©  the  quotient 
figure  smaller. 

When  tbe  num.         .  When  the  given  num- 

ber is  not  a  per- ber  is  not  a  perfect  square  annex  ciphers  for  new 

^  periods. 

When  the  trial     The  quotient  figure  will  be  a  cipher  when  the  trial 
divXnd''"^'^'^^^''*^^'^^''  '^  ^i'^-'ater  than  its  dividend. 

3.  AVIiat  is  the  square  root  of  3600^ 

4.  .What  is  the  square  root  of  3^^0625  ? 

5.  What  is  the  square  root  of  5764801  ? 

6.  What  is  the  length  of  one  side  of  a  lawn  which 
cont.'jns»Oi  acres,  or  400  rods,  if  made  into  a  square? 

7.  What  is  the  square  root  of  15025  ? 

8.  If  a  square  field  contains  ^400  sl][uare  rods,, what 
is  the  measurement  on  each  side?  Ans.SO. 

9.  What  is  the  square  root  of  15G.7325? 

A  number  both     REMARK. — A  ntlniber,  partly  integral  and  partly 

mtegrfiiandde-jecinKvl,  is  extracted  in  the  same  way  as  a  whole 

number;  the  first  point,  in  such  case,  must  be  placed 

over   the   units    and    extend    both    right    and   left; 

thus,  156.7325. 

The  num>,er  of     The  number  of  integral  figures  in  the  root  is  as 
Inu^'roof^^^^^^^y  ^^  there  are  periods  of  integral  figui'es  in  the 
power;  and  for  each  period  of  decimals  in  the  jiow.er, 
there  Will  be  a  decimal  figure  in  the  root. 

10.  What  is  the  square  root  of  3371.4207  ? 

4.ns.  57.19.  • 

11.  What  is  the  square  root  of  .25? 

12.  What  is  the  square  root  of  1.44?       Aiu.  1.2. 

13.  What  is  the  square  root  of  1  ?  Am.  i, 


APPLICATIONS  'IN  SQUARE  ROOT.  170 

Eemarkv — We  reducG  a  Vulgar  fraction  to  its  sim- To  find  the  root 
plest  form,  and  then  take  the  root  of  tl\e  numerator  ;',|"„j\''"'s'*'^^'"^° 
and  denominator  gopai'atoly. 

14.  W hat  is  tttq  square  root  of  30}  ? 

.  *        '  '     ' 

Eemark. — When  either  term,  after  being  reduced, .when  either 
is  an  imperfect  square, -vvo  change  the  fraction  to  a  Jforfect, s^qiwre" 
decimal,  and  proceed  by  directions  already  given. 

15.  What  is  tlic  square  root  of  i?         Afis.  .8Gi). 

16.  What  is  the  square 'root  of  i-J^?  Ans.  f. 

17.  W!i;;f  is  nio  (square  root  of  i  +  f+l—- iV? 

v/l+l+i-T^6=n=f=ll.  Ans. 

18.  W!i:ii  i..  ..lo  ditYcreuce  between  v/9  and  9"  ? 

19.  What  is  the  difference  between  v^lG  and  •v/9  ? 

20.  Wiuit  is  the  ditferchce  hetweern  -v/g^-  and  P  ? 

21.  What  is  the  dilfereuce  between  ^/4  and  s/Q  ? 

,    '  '    '   ■  .  Ans.  5.    ■ 

22.  ^bat  is  the  sum  of  >/30i  and  272i  ? 
28.  Wlia.t  is  I  he  square  root  of  n/'980}  ?    , 

24.  What  is  the  dift'erence  between  v/81  aud  8P  '^ 


Ar'PLTCATlONS  IN  SQUARE  KOOT. 

ui;f[nition.^. 

I  i        332.    A   square   is   a  figure   witli  Definition  of 

I  ij  four    equal    siJcs.    and    four    equal  •'^'^'"^''°- 

H  ,  li  iuiirlcs,  the  angles  being  where  the 

B  R  bidea  jjieet. 


180 


APPLICATIONS   IN    SQUARE    ROOT. 


The  vertex. 
^  right  angle. 


A  parallelo- 
gram. 


333.  The  point  of  meeting  is  the  vertex. 

334.  The  sides  or  lines  of  a  square  being  perpen- 
dicular to  each  other,  make  each  angle  a  right  angle. 

Example  1. — If  an  acre  of  land  be  laid  out  in  a 

square  form,  what  will  be  the  length  of  each  side  in 

rods?  1x4=4x40=160.  ^ns. 

A  parallelogram  is  a  figure  which  has  its  opposite 

.,   sides  of  equtil  length,  and 

its  opposite  angles   equal. 

In    the    figure,   two   para- 

lellograms    are    described. 


A  triangle. 


To  find  area  of 
an  irregular 
field. 


2.  1)1  a  room  16  feet  long  and  11  feet  wide,  how 
many  square  feet  are  there?  Ans.  176  feet. 

A  triangle  is  a  plain  figure  of  three 
sides  and  three  angles. 

The  area  of  an  irregular  field  is 
found  by  its  being  marked  into  tri- 
angles. 


3.  What  is  the  area  of  a  tria^ngular  piece  of  land, 
one  side  of  which  is  40  rods,  and  the  distance  from 
the  corner  opposite  that  side  to  the  other,  20  rods  ? 

A71S.  -2/x 40=400  rods. 
Definitions  of  a     A  right  angled  triangle  has  one  of  its  angles  a 
Mgie^aud'  its'  right  angle.     The  side  opposite  the  right  angle  is  the 

hypothenuse,  the  lower  line 
the  base,  the  other  the  per- 
pendicular. 

lu  a  right  angled  triangle, 
the  square  of  the  hypothenuse 
is  equal  to  the  sum  of  the 
squares  of  the  other  two  sides. 
This  is  practically  apparent  by 
the  following  diagram,  in  which 
the  small  squares  on  the  hy- 
pothenuse, 25=  those  of  the 
base,  16+ those  of  the  perpen- 
dicular, 9=25. 


parts. 


What  the 
square  of  the 
hypothenuse 
equals. 


Base. 


APPLICATIONS  IN  SQUARE  ROOT. 


181 


/ 


4.     In      a     vigllt  Diacram  and 

angled      triangle,  ^^^iCJI^^^' 
Avhosc   base    is    4  sq"='re  of  iiy- 
yjirus     and     per- 
pcnclieiilai'8,  \;;hat 
is    tlio    hypotlicn- 
usc  i' 


<X^ 

u 

^v 

y 

y              BASE  4 

ccrlaiii      «laLv    is 
(lliC     KOllth     filt}'' 

loni^ucs,  the  other 
due    west    forty 
loiigucs:  how  I'ar. 
nl  that  time,  were 

they  aj  wvt'i 

OPEUATION. 

45=16,  and  o* 

=9;  thenv'i^+Ki 
=5,  the  hypoih- 
Cnusc. 

5.  Two  ships 
Bail  from  Charles- 
ton; the  one  at  a 


Note. — Find  the  hj'pothenuse 

Remark — When  the  base  and  perpendicular  are  to  find  hypoth- 
known,  we  tind  the  hypothenuse   by  squariiip;  tlie  ''''"''®- 
base  and  perpendicular,  adding  their  results,  and  ex- 
tl'acting  the  squire  of  the  sum. 

0.  There  is  a  street,  in  the  middle  of  which,  if  a    . 
ladder  50  feet  long  be  placed,  it  will  reach  a  window 
36  feet  from  tlie  ground,  on  either  side  of  the  street : 
how  wide  is  (he  street?  *" 

Eemark. — Having  squared  the  hypothenuse  and  wimn  the  hy- 
tho  known  side,  the  square  root  of  the  dltference  1^'^'^^''^^^ 
will  be  the  otlier  side.  known. 

7.  What  is  the  height  of  a  hou'c  which  .is  reached 
\)y  a  ladder  >  *  foot  long,  striding  in  a  sLroot  SJ  feet 
wide  ? 


APPLICATIONS    IN  -SQUARE   EOOT. 


i('S  eiieumfer- 
cnce. 

TJ)e  divisions 
of  a  cirjuirlfor- 
«nce. 


•1    (-•liorrt. 


A  diamrt<>r. 


A  rndius. 


4  tangent. 


8.  If  a  line  125  feet  long  will  reach  from  the  top  of 
a  tower,  100  feet  high,  to  the  opposite  side  of  the 
street,  what  is  the  width  of  the  street  ? 

9.  What  is  the  height  of  a  pine,  tlie  line  of  the 
hjpothenuse  being  100  feet,  and  from  the  foot  of 
hypothenase  to  the  base  of  the  tree  f)7  teet  !* 

33-:?,  A  circle  is  a  plain  figure  whoso  c'.en  trc  is  every- 
:"'•-.■.-.   V-..  ._„^„^,- ^^'■'^^^''^  equally  distant 
y'""^^   i     "'■•<  '   '  from     tlie     bounding 

curve,  called  the  cir- 
cumference. 

S36.  The  circumfer- 
ence is  cliYidod  into 
36ii  equal  parts,  called 
degrees,,,  and  desig- 
nated by  a  small  -ci- 
pher slightly'  elevated 
at  the  right  of  the 
number;  thus,  360°. 
The  semi-circumference  is  this  equally  divided,  or 
180°;  the  quadrant  one  fourth,  or  9<J°  ;  the  sextant 
one  sixth,  or  60°;  the  octant  one  eighth,  or  45°,  and 

o."j.  Any  portion  of  the  circamference.  say  C  I)  E,  . 
is  an  arc. 

338.  The  straight  line,  C  E,  connecting  the  ex- 
tremities of  an  ate,  i's  a  chord. 

3SJ>.  The  line  which  passes  through  the  centre  of 
a  circle  is  :i  diameter,  as  B  E.  ' 

34€;.  A  straight  line  from  the  centre  to  the  circum- 
ferenvv  .s  a  rridius,  as  G  A,  G  C,  G  E       . 

341.  A  straii!;ht  line,  as  A  E,  which  touches  the 
cifciiiiifci'cnce  m  one  point,-A,  and  cannot  touch  it 
elsewhei'e,  is  a  tangent.  ■ 

ExA^ViPLK  I. — What  is  the  area  of  a  circle  whose 
diameter  is  6  feet  and  circumference  1'.)  feet  ? 

'  ■    ■      ,  Ans.  2Si  feet. 


To  find  area  of 
a  circle. 


uPKRATION. 

i  Diameter 

•^  Circumference 


Or     r„   Ji592 
9 

'  t8":i7432§=328i. 


=3 

27 


EXPLANATION. 

We  here,  to  find  the 
area,  multiply-^-  the  di- 
ameter by  ^  the. circum- 
ference, or  we  square  i 
tlie  diameter  and  multi- 
ply it  by  3.141592dec. 


CUBE   ROOT. 


18.^ 


3.  What  is  the  area  of  a  circle  whose  diameter  is 
^Ofeet?  ,  ^«.s.  814.15. 

3.  What  is  the  area  of  a  circle  whose  dianiete];'  is 
•J8  feet  -( 

4.  WJiat  is  the  area  of  a  circle  whose  circiinifer- 
-nce  is  314.1592? 

B14.1592-=-3.1 11 5923=100,  tbo-  diamcL  .  7  lis 
(which  is  \  3.U1592)+100'^=7853.98,  the  area.   Aas. 

5.  The  dfiameter  of  a  circle  is  25  :  what  i^  the  side 
>f  the  inscribed'  sqiuu'O  ? 

x/^l'  V31.2.5=^i7.G77.  Ans. 

0.  The  diameter  of  a  circle  is  36 :  what  fs  the  side 
>f  the  inscribed  square?  ,     ..  .  . 

7.  The  circumference  of  a  circle. is  314.1592  :  whnt 
K  the  side  of  the  inscribed  square?  An^-. 


CUBE    EOOT. 

342,  A  Cube  is  a  solidity  of  six  equal  sidea^  aud  each  a  cui- 
)f  these  is  an  exact  square,      i      •  . 

343'.  The  cube  root  of  a  number  is  that  one  'v.iiich, -^cuber.->ot 
multiplied  into  itself  three  times,  produces  tlu-  i      ^  ,■,• 
\'hose  cube  is  to  be  evolved.      >'■    ' 

344,  The  numbers  in    the  first  line  of  .1... 
)rder  are  the  cube  roots  of  the  corresponding  on 
■i^ocond.     The  last  are  perl'ect  cubes. 

1,  2,  .3,    4,      5,      6,      7,      8,      !). 

1,  8,  27,  64,  125,  216,  343,  512,  72'J 

..  EiKAMPLB  1. — What  is   the  length  of .  each 

cubical  bli.cl 
inir  1'1\)  oi'i 


>,  !:,^-  Cube  roois  anif 
,  1  "  perfect  cube:^. 

ni  the 

Their  Humeri- 


t- 

^     ;         ;         ; 

CT 

1     .    j         I 

'•« 

S 

u 

UJ 

X 

UCMCTH  .q  FT 


!iXi>x9-  ■ 
X81=7J9; 
is  the  numbi 

X,,lr. Tr 

cube  is  to  lin 
bor  which,  n. 
into  it.sclf  ll. 
or  njultipliti 
its  'Squ-iro, 
the  given   n  i 


•  I  To  find  a  oabo. 
:i..!ied 


184  CUBE   ROOT. 

« 

2.  What  is  tlio  length  of  each  side  of  a  cubical  block, 
containiiii;-  lUiJU  culiic  inches?  Ans.  10. 

8.  Wlutt  i.s  the  cube  root  of  21024576  ?       Ans.'ZlQ. 

,    ♦  .  OPERATION. 

To  extract  the  12102-1576 

cuberoot.  1st  liial  4ivisor=20'^ x3=1200^  8 

20x7x>]=  420 ' 

7^=     49 

1069  1.S024  1st  dividend. 
2d  trial  d"        ..  '''Xo=2187U0  11683 

1^70X6X3=     4860 

e»=      36""^ 


LM  tuw  uivis()r=223596 


1341 576  2d  dividend. 
1341576  , 


KXPLANATION. 

First,  We  separate  into  periods  of  3  figures  each,  the 
number,  placiuL;  a  point  over  units,  thousands,  etc. 

JSecohd,  VYe  find  by  trial  the  greatest  cube  in  the  left 
hand  period,  and  jilacing  its  root  as  in  square  root,  sub- 
tract the  culic  v8«>  from  fhc  left  hand  period,  and  to  the 
.   remainder  annex  the  next  periocl  for  a  dividend. 

2Viir(l,  AVe  s(|u:ire  the  root  figure,  and,  anoesing  two 
ciphers,  multiply  this  result  by  3  for  a  trial  divisor; 
then  we  divide  ihe  dividend  by  the  trial  divi.-or,  and  set 
the  quolicnt  as  (he  f-ecoi.d  figure  <jf  the  root. 

Fvuri'h,  \ye  multiply  the  second  root. figure  by  the 
first,  annex  one  cipher,  and  multiply  thi.s  result  by  3; 
thou,  adding  the  last  product  and  the  square  of  the  last 
root  figure  to  tiie  trial  divisor,  we  have  in  the  sum  Che 
true  divisor. 

Fifth,  We  multiply  the  true  divisor  by  the  last  root 
figure,  subtract  tlie  ])roduet  from  the  dividend,  and  to  the 
.  remainder  annex  the  next  period  for  a  new  dividend. 

Sixth,  We  find  a  new  trial  divisor  and  proceed  as 
before  until  all  the  periods  have  been  used. 

The  true  figure      Note. — The  true  figure  can  never  exceed  9,  and  lias  to 
never  above  9.  ^jg  invariably  ibuiid  by  trial. 

4.  What  is  the  cube  root  of  17576?  Ans.  26. 

5.  What  is  the  cube  root  oi  97a-'99  ?  A^v-.  99. 

6.  What  is  the  cube  root  of  3700416?         Ans.  156. 


CUBE   ROOT.  18& 

7.  What  is  the  cube  root  of  ^^84.604519?  .4ns.  4.39. 

8.  What  is  the  cube  root  of  74.088  ? 

Remark. — To  extract  the  cube  root  of  a  vulgar  frac-To  extract  the 
tion,  let  the  fraction  first  be  reduced  to  its  lowest  terms,  ^Jj{'®'^°^^^^^^jj* 
and  then  the  cube  root  of  the  numerator  and  denominator 
be  extracted   separately,  if  exact ;    if  they  are    not,   let 
ihem  be  reduced  to  a  decimal  and  then  take  their  root. 

9.  What  is  the  cube  root  of  -^^t 

»  Tiso  •- — v-'riis  —  s-    -i"^^' 

•    10.  What  is  the  cube  root  of  ilV  ^"■'*-  f- 

11.  What  is  the  cube  root  of  3l-3'''4\  ?  Ans.  of. 

12.  What  is  the  cube  root  of  -^ '( 

3v/i=>^.2=58.  Ans. 

13.  What  is  the  cube  root  of  f^ '( 

14.  What  is  the  cube  root  of  f-J-jr  ? 

15.  What  is  the  cube  root  of  -^g't     ■ 
IG.  What  is  the  cube  root  of  of  ? 

17.  What  is  the  cube  root  of  -j-^^s '( 

315.  Example  1. — AVhat  must  be  the  length,  depth 
Aud  breadth  of  a  box  when  these  dimeuEions  arc  alike, 
and  the  box  contains  491;]  cubic  feet?  Ans.  17. 

2.  A  jeweller  has  two  small  golden  balls;  one  is  1  inch 
in  diameter  and  the  other  two  inches :  how  many  of  the 
smaller  will  it  take  to  make  the  larger  one  ?  •       Ans,  8. 

3.  If  a,  globe  of  gold,  ]  inch  in  diameter,  is  worth  §100, 
what  is  the  diameter  of  a  globe  worth  §7200  ?     Ans.  8. 

4.  If  the  diameter  of  the  sun  is  886.144  miles  and  that 
of  the  earth  7912  miles,  how  many  bodies  like  the  earth 

^will  make  one  as  large  as  the  sun?  Am.  1,404,928. 

5.  If  1000  bodies  like  the  earth  arc  required  to  make 
I  like  the  planet  Saturn,  and  if  the  diameter  of  that  planet 
is  79,000  miles,  what  is  the  diameter  of  the  earth  ? 

,        Am.  7900  miles. 
<5.  What  is  the  length  of  one  side  of  a  cubical  corn  bin 
that  contains  2500  buf^heb?  ^«.s.  14.58  busliels. 

7.  What  will  be  the  length  of  one  side  of  a  cubical 
block  which  contains  1725  solid  or  cubic  inches? 

Ans.  12. 
^.  What  will  be  -the  length  of  one  side  of  a  cubical 
block  of  granite,  whose  contents  shall  be  equal  to  another 
S2  feet  long.  16  feet  wide,  and  8  feet  thick? 

vV32x  16x8=16  feet.  Aiis. 
U 


166  MISCELLANEOUS    EXAMPLES. 


MISCELLANEOUS   EXAMPLES. 

346.  Example  1. — What  is  the-square  root  of  580644  ? 

Ans.  762. 
2.  x/ 1679 316=  how  many?  Ans    lii96. 

"  3.  How  many  figures  are  there  in  the  cube  of  99  ? 

Ans.  6. 

4.  How  many  figures  in  the  cube  of  40  ? 

5.  What  is  the  diflference  between  \K8  and  •v/4  ? 

6.  What  is  the  difference  between  -^ I. and  1'  ? 

7.  A  square  table  of  mosaic  contains  80,625  square 
stones  of  erjual  size :  what  number  is  in  one  of  its  sides  ? 

8.  A  person  wishes  to  form  a  tract  of  land,  containing 
50  acres,  2  roods  and  20  rods,  into  a  perfect  square  :  what 
will  be  the  length  of  each  side  ? 

9.  A  room  i^  25  feet  long,  20  feet  wide  and  12  feet 
high  :  what  number  of  square  feet  does  it  contain  ? 

10.  What  is  the  cube  root  of  ^^  ? 

11.  -^84.60 45 19= how  many?  Ans.  4.S9.     ■ 

12.  A  pipe,  f  of  an  inch  in  diameter,  will  fill  a  cibteru 
in  5  hours  :  in  what  time  will  a  pipe  'A\  inches  in  diameter 
fill  it?       .       .  Ans.  Saimin. 

13.;  What  is  the  cube  «root  of  28  ? 

14.  A  general  with  a  force  of  3136  men  wishes  to  form 
them  into  a  square :  how  many  must  he  place  in  rank  and 
file? 


MISCELLANEOUS     ARITHMETIC. 


TAIir    SIXTH. 


ARITriMKTK.'AIi    PROGRESSION. 


347.  A'-ithni-'M'ral  R-)/r.wifn  is  tho  torra  that  describes  Arithmeticnl 
an  incraa-iing  or  d3erja<iii4;  sories  of  numbers  by  the  addi-  ^[^ed'!'^^'"*^'* ^'^^ 
tioa  or  subtraetiun  of  th  ;  .sam  ;  ii  ;iire. 

348.  An  increiiiii;  sjries,.C)iJiiaonciiig  with  2,  is  2,  4,  An  increasing 
6,  8,   10,.  12,  U,  etc.  .  ««'■*««■ 

349.  A  decre-uiiij;  sorie;.  oiU'UcnciQg  with  14,  is,  by  A  decreasing 
the  subt-ictioDv  of  thj  8:itii;  cj.utujn  difference,  14,.  12, '^*'''*^'*" 

10,  8,  (-).  4,  2. 

350.  T'le   nuinSers  im  I   an;  tho  term^  of  the  series  or  Term«(: 
progression.     Tlij  first    an  I    List   are    the    extremes,  tTie  extr.-mes; 
others  tho  inaaHS.  "'*^*''"- 

351.  Ill  e.io!i  Arithin 'tical  Pry^rossion  are  five  parts  ;  jjumher  of 
three  of  whichbeiii  4  k.i  ),v.),  tho  two  others  can  be  found  :  ••'""^  »nd 

,,  ■^  '  '  numus. 

these  are, 

Ist.    Tho  first  term. 

2d.    The  last  term 

3d.     The  ciniUDn  dilVj-,'r»; 

4th.   Tnc  nnni'oer  of  terui^. 

5th.  Tho  sum  of  all  the  tc -.n.H. 

ExAM'r-L;'.  1. — .'V  Iv)y  Imu  .Ir  20  marbles,  and  paid  I 
cent  for  the  fir^t,  3  ce!»f<  fur  i  In.!  .^econd,  and  so  on  :  what 
number  of  ci'ias  did   ho  liu'0.1'  id  that  purchase  J* 

Alls.  39. 


188  ARITHMETICAL   PROGRESSION. 


OPERATION. 

To  find  any  dfin-      19  the  number,  less   1. 
ignated  term.  g  comuioii  difference. 


EXPLANATION. 

Here,  wo  multiply  the 
common  difference  (2)  by 
the   number  of  terms  pre- 

38  j   ceding  tlie  required  term; 
1  first  term.  I   we  then  add  the  product  to 

—  j   the    first    term    (the  scries 

39  la.st  term.  I  beiug  an  increasing  one), 
and  find  in  the  sum  the  answer  sought. 

Whpn  the  ae-       Note. — When  the  series  is  a  decreasing  one,  we  vary 
***"***"■  from  the  above  and  subtract  the  product  from  the  first 
term,  and  find  answer  in  the  diflcrcnee. 

2.  If  a  man  on  a  juurnoy  travel  the  fir,st  day  3^  miles, 
■  the  second  0,  and  so  ou  in  arithmetical  progression,  how 

far  will  he  travel  the  20th  day?  Aus:  f)lm. 

3.  A  man  liad  5  sons  who.se  ages  had  the  same  differ- 
ence; the  youngest  was  6  years  old  and  the  oldest  20: 
what  wa.s  the  common  difference  of  their  ages?    Ana.  4. 

4.  The  extremes  of  an  arithmetical  series  are  3  and  59, 
and  the  number  of  terms  8 :  what  is  the  common  differ- 
ence ?  Ans.  8. 

OPKIIATION.  ]  EXPLANATION.       , 

lommon  'differ-  59-?]=  5'.--t-7=  8.  |       The  extremes  and  number 

ence  wiien  ex- of  terms  having  been  given,  we  found  the  common  differ- 
number  of  encc  by  the  division  of  the  difference  of  the  extremes 
tenns  are  given.  ^^56^  by  the  number  of  terms  less  1  (8—1=7). 

5.  The  extremes  are  17  and  137,  and  the  number  of 
terms  9  :  what  is  the  common  difference  ? 

6.  The  extremes  arc  24  and  18U,  and  the  number  of 
terms  I'd  :  what  is  the  common  difforence? 

7.  A  man  had  6  sons  whose  several  ages  differed  alike ; 
the  youngest  was  o  years  old  and  the  eldest  28  :  wliat  was 
the  diflerence  of  their  ages  ? 

8.  A  man  has  10  sons  whose  ages  form  an  arithmetical 
series  ;  the  youngest  is  2  and  the  eldest  20  :  what  is  the 
difference  of  their  ages  ? 

9.  If  the  extremes  be  3  and  45,  and  the  common  differ- 
ence G,  what  is  the  number  of  terms?  Ans.  8. 

OPKKATION.  j  KXPLAKATTON. 

When  the  ex-  45— o=4l'-=-0=  7+1=8.  |  In  this  example,  the  ex- 
tremes and    ,  tromes  and  the  common  difference  beiit''  given,  we  divide 

Common  differ-    ,        ,.-,  ^.     i  .  /«-       •'_l^;)>    i        ii  „ 

uBce  are  given,  the  dinereiicc  of  the  extremes  /4n— v.=i«tj)  bv  tlie  com- 
mon difference  (6)+ 1=7. 


ARITHMETICAL   PROGRESSION.  189 

10.  A  man  being  asked  the  number  of  his  children, 
eaid  thr;  the  youngest  was  8  and  the  eldest  oG,  and  that 
the  increase  had  been  one  in  every  3  years  :  how  many 
had  he  ? 

11.  A  stone  falling  descends  ]  6  J  feet  in  the  first  second, 
and  209 1^  in  the  last  second ;  the  increase  of  its  velocity 
a  second  being  32  i  feot :  in  how  many  seconds  docs  it 
fall  ?  Ann.  7. 

12.  How  many  times  does  a  clock  strike  in  12  hoi^rs? 

Ans.  78. 

OPERA^TIOX.  1  •       EXPLANATION'.. 

1+12=18x0=78.         I       We  multiply  the  sum  of  when  the  ex- 
the  extremes  (L+12)  by  one  half  (=6)  the  number  of  *''''"^r  *"^ 

^     '        /      J  V         y  number  of 

terms.  terms  are 

13.  If  a  piece  of  land,  60  rods  in  length,  be  20  rods  ^'^on. 
wide  at  one  end,  and  at  the  other  terminates  angularly, 
what  is  its  number  of  square  rods?  Aiii^.  600. 

14.  The  first  term  is  5,  the  common  difference  8,  and 
the  number  of  terms  21  :  what  is  the  sum  of  the  series  ? 

Am.  1685. 

OPERATION.  EXPLANATION. 

8x21=168  We  find,  in  this  case,  the  when  the  first 

Add  5  last    term    by    multiplyins^'^U"'  <^"'"'"on 

the  common  difference  (8)  number  of 

1685  by   the   number    of   terms  *^'^'""*'^  given. 

(21)  and  adding  first  term  to  the  product. 

15.  The  first  term  is  7,  the  common  difference  9,  and 
the  number  of  terms  25  :  what  is  the  sum  of  the  series  ? 

16.  The  first  term  is  f,  the  common  difference  3  J,  and 
the  number  of  terms  65  :  what  is  the  sum  of  the  series  ? 

17.  A  falling  body  descends  16-i\-  feet  in  the  first  second 
of  time,  and  the  increase  of  velocity  is  32^  feet  each  suc- 
ceeding second  :  how  far  will  it  fall  in  8  seconds  ? 

18.  The  sum  of  an  arithmetical  series  is  7,  the  number 
of  terms  8,  and  the  least  term  4  :  what  is  the  greatest 
term?  Ans.  14. 

OPERATION.  I       EXPLANATION. 

>  8-h2=4  ;  72-4=18,  and  18-4=14.|  We  divide  the  when  the  sum 
sum  of  the  series  (72)  by  half  the  numbtir  of  terms  (4),  ofaseriea.num- 
and  subtract  the  gi.ven  extreme  (4)  for  answer.  and  one  ™^ 

19.  The  sum  of  a  series  is  204,  the  number  of  terms  ^^^^^j®  *^® 
12,  and  the  greatest  term  is  39  :  what  is  the  least  term  ? 

20.  A  falling  body  descends  1029 J  feet  in  8  seconds; 
in  the  8th  second  it  falls  24H  :  how  far  does  it  fall  in  the 
first  second  ?  Am.  16-iV- 


190  GEOMETRICAL   rROGRESSION. 


GEOMETRICAL  PROGllESSION. 

Goomctrical  352,    Coometriral  Progression  is  the  t-onu  applied  to  that 

nnd.  part   or  arithmetic  which  shows  the  increase,  by  niultipli- 

cation,  or  decrease  by  division,  of  a  uuiuerical  series  through 

a  common  number. 
Tiie  ttrms,  353    i\yQ  Duuibers  of  these  sefics  are  terms ;  the  first 

and  last  are  the  extreuie ;  the  otliers  uican. 

The  ratio;  354.  The  coUiUiou  number  employed  is  the  ratio.     It  is 

when  an  into- g^  intotrer  when  the  series  increases,  but  a  fraction  when  it 
ger ;   wlien  a      ,  '^  ' 

Traction.  dccrca.se.s. 

An  increasing       365.  An  increasing  scries,  with  the  ratio  2  is,  1,  2,  4, 
cS;,/«en>8  ^»  l'^,  32,  G4,  1:^.8;  a  decreasing  .series,  with  the  ratio  |, 

^  is,  128,  64,  32,  10,  8,  4,  2,  1. 

Five  t«rmfl  and      366.  There  are  five  terms  in  Geometrical   Progression, 
thoirnamoB.     ^^^  three  of  which  known,  easily  determines  the  remain- 
ing two :  these  are, 

1st.  The  first  term. 

2d.    The  last  term. 

3d.    The  number  of  terms. 

4th.  The  sum  of  all  the  terms. 

5th.  The  ratio. 

Note. — By  the  ratio,  we  multiply  or  divide  to  form  the 
series. 

Remark. — A  proper  understanding  of  this  branch  of 
numbers,  requires  the  knowledge  of  Algebraic  equations 
and  logarithms.  In  this  work,  a  few  illustrative  examples 
only  are  given. 

Example  1. — The  first  term  is  3  and  the  ratio  2  :  what 
is  the  6th  term  ? 

OPERATION.  EXPLANATION. 

wiien  the  first     2x2x2x 2x 2=2*=32  We  multiply  the 

term,  ratio  and  ,         ^t    i 

number  of  dl.stterm.     first   term    by   that 

terms  arc  given.  rwiwpr    ni'    fbp    rntift 

to  find  the  lart                                           —  power   01    ine  raiio 

or  any  t«rm.                                              96  Ans.  whose  index  is  equal 

to  the  number  of  terms  preceding  the  required  term  for 

answer. 

Remark. — It  will  be  seen  that  the  last  term  is  equal  to 


PERMUTATION   AND   COMBINATION.  l'9'l 

the  first  term  multiplied  by  the  ratio,  raised  to  a  power 
less  1  than  the  number  of  terms. 

2.  The  first  term  of  a  decreasing  progression  is  12S,  the 
ratio  ^,  and  the  number  of  terms  7  :  what  is  ihe  last  term  ? 

(i/=A  128xA=V4«=ti.  An,. 

3.  A  sum  of  money  is  to  be  divided  ami.ng  !(•  persons; 
A  is  to  have  $10,  B  ?oO,  and  so  on:  what  will  the  tenth 
receive?  .  ^h.s.  8196,830. 

4.  A  lad  offered  to  purchase  17  oranges,  and  to  pay  for 
them  at  the  rate  of  one  cent  for  the  first,  2  cents  lor  the 
second,  and  soon  in  duplicate  order:  what  was  the  cost  of 
the  17t]i  orange  ? 

5.  The  cxtiemes  arc  -2  and  20,000,  and  the  ratio  10 : 
what  is  the  sum  of  the  series  ? 

OPEU.'VTION. 

20000—2=19998-;   10-1=9;    10998-^9=2222;    and  ,^;';;^,^^/^^ 
2222+20000=22222.  Aiis.  tio  have  bsoa 

civen. 
EXPLANATION. 

In  this  example  we  divide  the  difference  of  the  ex- 
tremes (101)98)  by  the  ratio  less  1  (10— 1=9),  and  add 
to  the  quotient  the  greater  extreme. 

6.  A  man  trading  for  a  horse  offered  to  pay  for  him  at 
.the  rate  of  a  cent  for  the  1st  nail  in  his  shoes,  8  for  the 

2d,  and  so  on;  there  were  32  nafls  :  what  ditl  the  horse 
cost?  ^H.s.  £9,205,100,944,  ;'.9.20. 

7.  A  father,  at  the  celebration  of  his  daughter's  birth- 
day in  January,  gave  her  §5,  and  said  he  would  double  it 
on  tho  first  day  of  each  successive  week  in  the  fifth  year: 
what  was  th(;  nina  that  he,  through  ignorance  of  Geometrical 
Progre.*sion,  pledged  himself  to  pay? 


PERMUTATION  AND   COMBINATION. 

357.  Permutation  is  a  process  to  find  the  different  ways  pcrmutaUfin 
in  which  numbers  or  things  can  be  placed;   Comhmation,^'^^  combin*- 
their  various  arrangements  in  sets  or  series. 

Example   1. — In   how   many   different   ways   can   we 
arrange  the  first  five  letters  of.  the  alphabet? 

Ans.  120. 


PERMUTATION    AND   COMBINATION. 


To  find  perm  ..-  ^ 

ut.ions.  1.  a  b  c  d  e 

2.  b  c  d  e  a 

o.  d  e  a  b  c 


OPERATION,  I  EXPLANATION. 

We  multiply  together  all 
tlie  terms  of  the  natural 
series   from    1    up    to   the 


4.  c  a  b  c  d,  etc.  j  given  number,  and  find,  in 

1X1'X3X4X5=120.  the  last  product,  the  uum- 

a    b     c     d    e=  5  letters,   i  ber  of  changes  sou2,ht. 


2.  How  many  different  integral  numbers  may  be  ex- 
pressed by  writing  once  in  each  number,  the  9  digits  in 
succession?  Ans.  1x2x3x4x5x6x7x8x9=362880. 

3.  The  solar  spectrum  consists  of  7  colors — red,  orange, 
yellow,  green,  blue,  indigo  and  violet:  in  what  varieties 
can  these  be  placed  ?  Ans.  5040 

4.  How  many  changes  can  be  rung  ou  St.  Michael's 
bells',  supposing  them  to  be  8,  and  allowing  3  seconds  to 
each  round  ?  .  . 

5.  How  many  different  companies,  each  of  7  men,  may 
be  selected  from  21  men  ? 

'   '  OPERATION. 

21x20x19x18x17x16x11?      ,,^,^^     , 
.^  ^---, — ~, — ,,—-„  =116280.  Am. 

Ix  2x  oX  4x  5x  6x  7 

EXPLANATION. 

To  find  the       •    Here   we  form  a  1st  series  of  numbers,  commencing 

bination.s'of'"  with  that  (21)  to  be  selected  from,  and  decreasing   as 

^^y  J^"""''^'' o^many  times  as  the  other  number  expresses;    and  a  2d 

series,  commencing  with  1,  increasing  to  the  number  to 

be  combined  for  a  divisor,  and  find  in  the  quotient  the 

combinatious  sought. 

6.  The  graduating  class  of  the  University  of  the  South 
consists  of  80  members,  12  of  whom  are  to  have  honors 
and  appointments :  how  many  different  selections  could 
be  made  ?      , 

7.  There  aje  said  to  be  56  different  elements  in  nature  : 
if  one  particle  in  each  element  will  combine  with  one  par* 
tide- in  c:'.ch  of  the  other  elements,  hovr  many  combina- 
tions niiVj  be  formed  ?  •         Ans.  1540. 


MENSURATION. 


MENSURATION. 


19i 


858.  Me.nmradon  is  that  part  of  arithmetical  science  S^ed*""" 
which  describes  capacities  of  various  kinds. 

Remark. — The  points,  lines  and   surfaces   named   in  Po'^'^s'' i'a<^?' 

.  ^     .      '  ,  .        ,  •  1  •    .  etc-,  not  roia. 

mensuration   arc    imaginary,  and  are,  simply,  aids   to   a 
mathematical  in([uiry. 

DEriNITIONS. 

^59.  Example  1. — A  point  has  neither  length,  breadth  A  point, 
nor  thickness,  but  only  position. 

2.  A  line  has  length,  but  no  breadth  or  thickness.  a  line. 

•- ,.     3-    A    right    line,    or    straight  a ^H,ht^.o^ 

line,  extends  only  in  one  direction. 

\  4.  A'  broken  line  is  formed'^  ^.roken  line 

\  _     of  two  or  f»ore  right  lines. 

^  5.    A   curved   line   is    one    tlmt    constantly^  '^""'^ 

changes  its  direction. 

j  G.  Two  lines'  are  perpendicular  to  each  P^^P^^'^'i'^'"''''' 
I other,  when  they  touch  so  as  to  form  right 

.1  angles. 

•  "  7.  Two  parallel  lines  are  everywhere  Parallel  lines, 

equally  distant  from  each  other.- 

8.  Two  lines  are  oblique  to  each  other  when  Obiiquo  linog. 
their   point  of  union   makes  acute  or  obtu.se 
auirles. 


0.  A  surface,  superficies,  or  area,  has  perfi<!ie^V" 
length  and  breadth,  but  no  thickness,  area, 
and  is  plane  or  curved. 

10.  A  plane  surface  is  such,  that  if  Apianesurfeee. 
any  two  points  are  assumed  upon  it,  the 
straight  line  joining  the  points  will  be 
wholly  upon  the  face. 
11.  A  curved  surface  constantly  changes  direction,  as  .\  curved  sur- 
the  exterior  of  a  globular  body. 


194 


MENSURATION. 


A  polygon.  ^o.  A  polygon  is  a  plane  fijjure  bounded  by  at  least 

throf  f^tiai^ht  lines. 
HoufXa.'''*'       '•^-   ^  Polygon  of  3  sides  is  cnlled'a  tviaogle ;  of  4,  a 

quadrangle ;  of  5,  a  pentagon ;  ol"  G,  a  hexagon ;  of  7,  a 

heptagon  ;  of  8,  an  octagon,  etc. 

14.  Tiian,i:!es  v'lh  3  equal  sides, 
are  equilatoral ;  Avilh  2,  are  isosceles; 
with  3  unequal  sides,  are  scalene. 

Kotc. — Fur  nicasurcimnt  of  tri- 
angles See  180th  page. 

360.  Example  1. — Vv'hat  is  th« 
area  of  a  rectangular  fit  Id,  whose 
leugth  is  40  ri'Jg  aud  breadth  20? 

Ans.  5a. 


■J  n'anglos,  equi- 
lateral ; 

i«08celes; 

scalene. 


To  find  the  arcn 
of  tt  parull<-,lo- 
gram,  rectwiglc 
or  hquare. 


OCr,KATION. 

4(» 
20 

100)800(5  acres 
800 


A  trapeaoid. 


EXri,.\NATIO.\. 

We  multiply 
the  base  by  the 
altitude,  and  haxe 
ihe  uii?wer,  800 
r(»ds.  This  re- 
duced by  the 
number  of  square  rods  (4x40)  in  a  square  acre,  gives  6. 
Note. — For  defiDitione  of  square,  etc  .  ^eo  Art.  382-334. 

2.  What  is  the  area  of  a  parallclngram  who.«e  base  is  2 
feet  and  height  3  inches?  Afin.  72  sq.  in. 

3.  What  are  the  contents  of  a  field  t'O  wkIs  .«(juare  ? 

4.  The  parallel  sides  ol'  a  trapezoid  are  7  and  11  feet, 
and  its  height  or  altitude  4  feet :  what  is  its  area? 

Ans.  3G  sq.  ft. 
Xolr. — ^A  trapezoid  is 
a  4  sided  fi;.;urc,  having 
two    of  its   sides   par- 
allel. 


\ 


OPERATION.  .    [  EXl'LANATION. 

Toflndthearea  ll+7=18x4=72-=-2=36.  i      Here  We  multiply  the  sum 
of  atrapezo.d.  ^^  ^j^^  ^^^  parallel  sides  (=18)  by  the  altitiide,  and  divide 
the  product  by  2,  which  gives  the  arca. 

5    What  is'tljc  area  of  a  trapezoid  whose  parallel  side* 
are  20.5  and  12.25.  and  its  altitude  10.75yds.  ? 

Ans.  1?6.031  sq.  in. 


MENSURATION. 


195 


Remark. — The  diagonal  of  a  trapezoid  divides  it  into  The  diagonal  or 
[ —  '~,-^\  *^wo    triangles    whose  *  """P"**"^" 

!  '      i  \  bases  are  tl)C  parallel 

i  -  \  sides  of  the  tmprzoid, 

j       ,,''■  ""\        and  whose  coiunioii  ill- 

i^tl' „.-j.     titude  is  the  altitude 

of  the  trapezoid. 

6.  What  is  the  area  of  a  trapezoid  whose  altitude  is  8, 
and  whose  parallel  sides  are  12  and  18?  Ans.   12i?. 

7.  What  is  the  area  of  a 
triangle  whose  base  is  60 
feet  an.d  altitude  8  feet  ? 
Ans.  200  sq.  ft. 

jVr/te. — For  definitions  of  a  triangle  see  Art.  358,  14  Dcf 


We  multiply  the  base  by '^P'^"'**'^''*'"^ 

^     •'  -^    of  11 


OPKRATIOV.  I  EXPLANATION'. 

50x1=  :oo. 

half  (8-r-2=  1)  the  altitude. 

8.  What  are  the  contents  of  a  triangular  field  whose 
base  iB  25  rods  and  altitude  18  rods? 

9.  The  sides  of  a  triuugle  are  8,  10  and  12  feet:  what 
is  its  area  ? 

Ot'EKATION. 

8+10+12=3f;-^:=i5;   15-8=7 
15-10=5j_l 5—  li'=3 ;  y/Uxfx 
5><y=1575=i0  sq    ft.  nearly. 


triangle. 


EXri.A  NATION. 

Imoui    the    '}  To  find  area 
!■      1        A.whon  three 
sum    ot    the    ff  .«Klesof  atrian- 
u_  gle  ore  known. 


sides  we  suD 
tract  each  .'sepa- 
rate side,  and  then,  by  multiplying  together  the  ^  sum 
(=15)  and  the  tiiree  reoiaiuders  (7,  5,  3),  we  have  in 
the  square  root  of  (ho  continued  product  the  are.i. 

10.  The  sides  of  a  triangle  are  7,  1.^  and  15  :  what  is 

its  area?  ^ns.v'l 700=41. 2o-t-. 

11.  The  sides  of  a  triangle  are  10,  15,  20  :  what  is  its 
area  ? 

12.  W^hat  is  the  area  of  a  parallelogram  whose  base  is 
30  feet  and  altitude  8  feet :  and  what,  if  diagonally  marked, 
would  be  the  areas  each  of  the  triangles  formed  by  such 
lino  ? 


196 


MENSURATION. 


The  diagonni  of 
a  purallelo- 
yram. 


Remark. — The  diagonal  of  a  parallelogram  separattf 

it  into  2  equal  tri- 
angle's, whose  bases 
I  aud  altitudes  art- 
each  equal  to  th«' 
base  and  altitude  of 
the  parallelogram. 

KXPLAXATION. 

'Jlio  ba.se,  multiplied  by 
altitude,  gives  the  first  an- 


OPKRATION. 

24()-j-2=120. 


To  hnd  lirra  of 
a  parallelo- 
gram; also.  •  1  o- • 

wh«n  triangied.  swer ;  that,  divided  by  2  (=f),  the  Bccoud. 


Note.. — The  area  of  a  triangle  is   I   the  area  of  a  paral- 
lelogram that  hat»  the  yMxu-  base  and  altitude. 

l.'i.   What  is  tlu  J.  a  parallelogram  whose  base  if 

05  feet    and    altitude   10  ?    what,  if  d'agonally   marked, 
would  be  the  areas  of  its  triangular  divisions  ? 

1 1.  The  sides  of  a  square  are  10  feet  and  the  apothem 

6  feet:  what  is  its  area?  Aik.  100  sq.  ft. 


The     apotlicm 
of  a  polygon. 


Nott. — The  apothem 
of  a  polygon,  or  a  fig-' 
lire  with  many  angles, 
is  the  perjieudicuUr 
drawn  from  the  centre 
of  the  polygon  to 
the  middle  of  either 
side. 


I!  \T10X.  EXPLANATrO". 

To  find  .ire*  ot  10x4—      X -•K=^'''0=i'-'^-,      H<'-.e,    the    perimeter^    or 
any  regular  pui-|j^jy„j;„^  I  ^.u-s  of  the  figure,  ifi  multiplied  by  h  its  apothem. 
15.  Tbo  sides  of  a  square  are  30  feet  aud  the  apothem 
16  feet :  what  is  fts  area  ? 

The  area.s  <if  all  similar  figure.-?  are  to  each  other  as  tht 
squares  ol  their  homologous  or  proportionat<^  sides.  By 
the  following  table  of  the  areas  uf  regular  polygons,  when 
each  side  is  a  unit,  we  can  easily  calculate  the  areas  of 
fisnires  therein'namcd  : 


MENSURATION. 


197 


Nil  me. 


|No.  Sides        Apothem. 


Trianiilo ?>  \  0.2886751 

Square  4  0.5000000 

Pentatron 5  1  0.G881910 

Hexairou 6  |  0.8660254 

Heptagon 7  j  1.0882607 

Octagon  8  j  1.2071068 

Konagon..; 9  i  1.37^7387 

Decagon .1  10  {  1.5388418 

Tlndeeagon...'-.....!  11  |  1.7028436 

Opdecagon 12  I  1.8660254 


Area. 


0.43y0127 
1.0000000 
1.7204774 
2.5980762 
3.6339124 
4.8284271 
6.1818242 
7.6042088 
9.365G899 
11.1961524 


A  table  for  facil- 
itnnngarealCBl- 
cul.itions  of 
polygon.s. 


16.  The  side  of  an  octagon  is  10  feet  and  its  apotliem 
12.07106  :  what  is  its  area  ?  Am.  482.8427  sq.  in. 

Ol'EKATION.  I    EXPLANATION. 

10x10=100x4.8284271=482.842-1-'.  I      Wc    square  to  find ftr**  of 
one  side  of  the  polygon  whose  area  is  required,  and  mul- »  r'"'.vgf>n  by 
tiply  the  square  by  the  tabular  area  of  the  polygon  iiamed,    '', 
for  area. 

17.  What  is  the  area  of  a  regular  triangle,  one  side 
'>eing  8  inches  ?  . 

18.  What  is. the  area  of  a  square,  one  side  being  12 
inches? 

19.  What  is  the  area  of  a  heptagon,  one  side  being  3 
■feet?   ,  • 

20.  What  is  the  area  of  an  octagon,  one  of  the  sides 
measuring  8  rods  ? 

21.  What  is  the  circumference  of  a  circle  whose  diame-Tofipd  the  cir- 

.     r       •\      'i  I         1  -  -/io<v      -1  oumference  of 

er  IS  5  miles  .'^  Ans.  lo.iuSO  railcs.      ^circio. 

OPEKATIOX.  '  EXPLANATION. 

5X3.1416=15.7080.        |     The  diameter  is  multiplied 
by  3.1410dec.,  and  gives  very  nearly  the  circumference. 

Note. — For  definitions  of  a  circle  and  its  divisions  sec 
\rts.  335-341. 

22.  What  is  the  ciroumfercnce  of  a  circ>e  whose  diame- 
ter is  30  feet  ?  Ans.  94.2480  feet. 


iVo^c— The  diameter  of  a  circle  is  found  by  the  inverse  J/;, Jt°?orac?^" 
operation  :  94.2480-t-3.1416=30.  cie.  ■ 


198 


MENSURATION. 


23.  What  is  the  circumference  of  a  circle  whose  diame- 
ter is  20  miles? 

24.  What  is  the  area  of  a  circle  ivhose  diameter  is  7  ? 

Ans.  38.4846. 

OPERATION.  I  EXPLANATION. 

Toftnd  the  area  7x7=40x7854=38.4846.  |     Multiply  the  scjuare  of  the 

?h'e^tt^r"« diameter  by  the  decimal  .7854. 

given.  25.  What  is  the  area  of  a  circle  whose  diameter  is  9/ 

Ans.  (58.6174. 

26.  What  is  the  area  of  a  circle  whose  diameter  is  20 
miles  ? 

27.  The  diameter  of  a  circle  is  20  feet :  what  is  the 
side  of  the  inscribed  square?  Ans.  14.142ft 

Ol'EllATIOX.  KXI'LANATION.      ■ 

20'=40U  ;  400-4-2=200  ;        We    extract    the   square 
14.142.  root  of  i  of  the  square  of 

the  diameter. 

28.  What  is  the  side  of 
the  greatest  square  stick  of 
timber  that  can  be  hewn 
from  a  cylindrical  log  36 
inches  in  diameter  ? 

29.  What  is  the  convex 
surface  of  a  prism  whose 
base  is  bounded  by  7  equal 
sides,  each  being  33  feet, 
and  the  altitude  22  ? 

^Hs.  6082  sq.  ft. 


To  find  the  side 

of  a  square  in-  i        ,r.r. 

Bcnbecf  10  a  cir-    <*"'J  v'-'^'-' 

cle. 


A  piiim  de- 
fined. 


Remaek. — A  prifm  is  a  solid*  with  two  pimilar  equal 

parallel  ftices,  called 
bases,  and  its  other 
laces  parallelograms. 


OPKRATION.  I  EXPLANATION. 

Tofindthecon-     33x7=231x22=5082.     1     Here   multiply   the    pen- 
rnghrpnsm.°^nietcr  of  base  by  the  altitude. 

Whenitisright;      Notc.—K  prism  is  Called  right  when  its  edges  are  per- 
when  oblique;  pg^tiieular  to  its  bases  J  when  not.  perpendicular,  oblique. 
terfotc."*"*^'''  It  is  also  triangular,  quadrangular,  etc.,  according  to  its 
bases. 


*  A  solid  is  a  figure  which  has  length,  breadth  and  thickneee. 


MKNSURATION. 


199 


30.  What  is  the  convex  surface  of  a  prism  when  there 
are  6  equal  sides,  each  IH  inches  ia  length,  and  in  altitude 
14  inches  ? 

Kofe. — When  a  prism  is  bounded  ^r**"*"®'"?'?^ 
G    parallelograms,   it    is    called   a 


p;iralIelopipedon. 

Whon   these   6  parallelograms  are  When  it 
n-ctaiigles,    the     parallolt>|)i|>e(l()n     jj- '""°"'*'^'' 


rrr'tangnlar  ;    when   equal   rcctaiigles,  wheu  a  cube. 
it  is  a  cube. 

•U.   What  is  the  convex  .surface  of 

!i   prism  with  6  e(]UHl  sides,  ouch  14 

inches  long,  and  the  altitude '•>  inches  ? 

32.  What  is  the  cnnvex  surface  of  a  cylinder  whose 

altitude  is  50  foet  and  the  diameter  of  whose  base  is  20 

feet  ?  Ans:  8  U I  .o  ^q.  ft. 

J^ote. — A  cylinder  is  a  round  body,  whose  diameter  is  ^  (cylinder  d«- 


\ 

k 

^ 

1---  ■-■ 

4 

unvariable,  and  whcse 
ends  are  equal  and 
parallel  circ!<s. 


62«320 

50  altitude. 


OrERATION'.  EXPLAVArroX. 

3.141()    docim-il.  Here,  as  we.  did   to  find  To  find  convci 

20  diameter.  the   diameter   of    a   circle,  f.'^'j^^f  °^  "  ^^' 

Art.  361),  22  F<x.,  we  mul- 
tiply .3.1410  by  the  diame- 
ter, and  that  result  by  the 
altitude. 
8141.6000 

33.  What  is  the  Convex  surfice  of  a  cylinder  whose 
altitude  is  1  foot,  and  the  circumference  of  whoso  base  is 
1  foot  and  6  inches!'  Anx.   21(i  sq.  in. 

34.  What  are  the  cmtents  of  a  cylinder  whose  base 
diameter  is  14  and  alr.itn  ie  25  ?  .4«s-.  ;>,S-t>>  4. 

OI'UKAilnM.  IkXI'LAN  AIION 

14xl4=196x.7S5t=15:}9384x25=3848  4l       In      this  to  find  the  con- 
case,  having  found  the  arci  of  the  base  by  nmitiplying  t,he  ^^"/f '^  * '^y'"*' 
square  of  14  by  decimal  .7S54,  we  find  the  coiiicnts  by 
multiplying  that  result  by  the  altitude. 

35.  What  are  the  ontents  of  a  cylinder  wlhc^o  base 
diaraet«^r  is  1^  and  wKdsc  u'titutle  is  'i'' 

36.  How  miny  incbc'*  in  a  cylinder  whose  b.isc  di.iuiL^tcr 
is  15  and  whose  altituJe  is  6  f 


300 


MENSURATION. 


37.  What  are  the  contents  of  a  pyramid  whose  area 
base  is  60  and  altitude  18  ?    '  Ans^  360. 

A  pyramid  de-  JVote. — A   pyramid    is    a  'solid 

^'*®^'  ''^v  having   a   polygonal  face,   called 

the  base.  Its  other  faces  are 
called  triangles,  and  meet  at  a 
common  point,  the  vertex. 

OPER\TION. 

60xl8=1080-T-3=860. 

EXPLANATION. 

tenKf'a^^'ra"  """"^    °~"        ^        '^^^  ^^'^^  ^^"^  ^^  multiplied  by 

ml±°  a   yra  ^^^  altitude,  and  i  of  that  taken  for  the  contents. 

38.  What  are  the  contents  of  a  pyramid,  the  area  of 
whose  base  is  350  and  the  altitude  25  '( 

,39.  What  is  the  convex  surface  of  a  right  pyramid 
whose  *laut  height  is  28  feet,  and  the  cireuml'erence  of 
whose  base  is  42  feet  ?  Ans.  882  sq.  ft. 

OPERATION.  I  EXPLANATION. 

Tofind,thecon-  42x21=882.  |     The  circumference  of  the 

righrpyramfo!  ^'"^6  is  multiplied  by  ^  of  the  slant  height. 

40.  What  is  the  convex  surface  of  a  right  pyramid 
whose  slant  height  is  56  feet,  and  the  circumference  of 
whdse  base,  is  60ft.  ? 

41.  What  are  the  contents  of  a  cone  whose  altitude  is 
30  feet  and  the  diameter  of  whose  base  is  10ft.  ? 

Ans.  785.40  sq.  ft. 

A  „,,Do.  •  Note. — A  cone  is  a  pyramid  with 

circular  base. 

OrERATION. 

10  *X. 7854=785400x30= 
23562000-^8=785.40. 

EXPLA^ATION^ 

Here,  the  square  of  the  base 
multiplied  by  decimal  .7854,  and 
that  result  by  the  altitude,  with  a 
division  by  3,  give  the  contents. 

42.  What  are  the  contents  of  a  cone  whose  altitude' ia 
25  feet,  and  the  diameter  of  whose  base  is  8  feet? 

^ns.  418.88  sq.  ft. 

43.  What  is  the  convex  surface  of  a  right  cone  whase 
slant  height  is  90  feet  and  the  circumference  of  whose  base 
is  60  feet?  Atis.  27  sq!  ft. 


f-;i',N>iliIlAlW>-N  '■■ 

OPERATION.  ''         ]  EXPLANATION. 

B0x45=":J700.  i     We  multiply  the  circum- Tofindthocon- 

reuce  of  the  baso  by  ^r  thc^lant  height.  •  right  conf^" 

u4.  What  is  the  couvox  suri'ace  of  a  riijjht  cone,  whose 
4ant  height  is  oO  l^et  aud  bjise  circumference  18  feet? 

Xote. — A  cone  ii  called  right,  when  its  edges  are  per-  whmacono  : 
/•■ndicular  to  its  bases  ;  otherwise,  it  is  oblique.  oifliquel'^ 

i5.  What  is  tlie  surface  «'>f  a  *ephere  whoae  diameter  is 

■Am.  201. OG. 

j'Vw.' V  o|'herc  is  asolid  body  a 

Nunded  by  a  curved  surface,  all 

;i:irfcs  bfting  equi-distant  from    a 

A      poiut   within,  called   the   ceutre. 

"' -i  '  diameter,  is   a  straight   line  hn  dnmct^r. 

•ssing  through  its  centre.  Ail 
diameters  6f  the  same  sphere  are 
equal'.  '     "  ^' 

Rkmark. — The  surfrice  of  :i  sphere  equals  4  great 
•circles  of  the  same  sphere  j  and,  as  we  find  the.  area  of  a 
circle  by  multiplying  (Ex.  1,  App.'  Sq.  11.)  the  circum- 
ference by  \  dianietei',  so  we  find  a  spliQrical  surface,  as 
shown  in  the  explication. 

OPKRATTOX.  1  EXPLANATION.  ^ 

8'*:=^64x3.14U)^-:^:i01.UG.  j  in  this  case  we  square  the  to  fi,jd  the^nr 
diameter,  and,  multiplying  it  by  3.]41Gdec.,  find  the  *""•=  jf a^P^^en- 
surface.  ■" 

46.  What  is  the  Rrirface  of  a  sphere  whose  diameter  is 
Oft.  ?  '  Am.  254.46  sq.  ft. 

47.  If  the  sun's  diameter  is  896,000  miles,  what  is  its 
surface  ?  Ans.  2,522,120,323,072  sq.  m. 

4S.  Whot  arc  the- contents  of  a  sphere  whose  diameter 
is  T)!';  y  Aiis.  65.450  sq.  ft. 

OPERATION.  1  BXI'LANATION. 

5 ""  X  3. 1416=78.540 X  5=]  .     AVe  multiply  the  surfiice  to  n^rf  the c(m- 
.'10:^.700-=-6=65.450.  by  the  diameter,  and  divi.do  t-mts  or  sphere 

!   that  by  6. 

40.  What  are  the  contents  of  a  sphere  whose  diametcc 
is  20  feety  ■      '■  "■     <  '         ^ 

50.  What  arc  the  coatents  of  a  sphere  who?36  diameter 
i^  28  rods'!' 

51.  What  are  the  contents  of  a  sphere  whose  diameter 
is  8000  miles? 

15  ■ 


202  ♦        MISCELLANEOUS    EXAMPLES. 


MISCELLANEOUS    EXAMPLES. 

361.  Example  1. — A  merchant  bought  12  cases  of 
merchandise  for  $669 :  what  would  25  have  cost  at  the 
same  rate  ? 

2.  If  8yds.  of  cloth  cost  £20  18s.  5d.,  what  is  that  per 
yard  ? 

3.  If  9f  barrels  of  flour  cost  £21  3s.  8d.,  what  would 
17-f  cost  ?  •  ■ 

4.  If  f  f f  a  ship  is  worth  £865  5s.  9d.,  what  is  the 
whole  worth  ? 

5.  What  number,  multiplied  by  J  of  itself,  will  pro- 
duce 7^  ? 

6.  What  number,  multiplied  by  i  of  itself,  will  pro- 
duce 5  J  ? 

7.  A  man  had  4cwt.  oqrs.  ISlbs.  of  tobacco,  which, 
equally  divided,  he  put  into  two  parcels  :  what  was  each 
division  ? 

51 

9.  If,  when  wheat  is  6s.  4d.  per. bushel,  the  penny  loaf 
weighs  8oz.,  what  ought  it  to  weigh  when  wheat  is  5s.  per 
bushel  ? 

10.  How  many  gallons  will  a  cisteru  contain,  the  diame- 
ter of  whose  base  is  10  feet  and  the  altitude  30  ? 

11.  If  a  staff  4  feet  long  cast  a  ahadow  of  6ft.  8in., 
what  is  the  height  of  a  column  which  casts  a  shadow  of 
155  feet  at  the  same  hour  ? 

12.  If  46  gallons  of  water  run  in  an  hour  into  a  cistern 
containing  21  h  gallons,  and  38  are  drawn  ofi  in  an  hour, 
in  wliat  time  will  it  be  filled  ? 

13.  What  is  the  square  root  of  9  times  the  square  of  16  ? 

14.  A  and  B  go  from  the  same  place  and  travel  the 
same  rosd ;  but  A  commences , his  journey  5  days  before 
B,  and  travels  at  the  rate  of  29  miles  a  day ;  B  follows  at 
the  rate  of  45  miles  a  day  :  in  how  many  days  will  B  over- 
take A  ?    • 

15.  IIow  many  cubic  feet  in  a  cistern  4ft.  2in.  long,  3ft. 
Sin.  wide,  ard  5i't.  7in.  high  ? 

16.  V/baf  is  the  square  root  of  the  square  root  of  ^g  of 
the  square  of  -,^a"s  ?  ^'^^-  A- 

17.  A  father,  in  his  will,  left  815,000;  of  this,  ^  was 
for   his   eldest  son,  \    to  the  second   son,  and  i  to  his 


8.  Multiply  -h  of  7-,^  by  ^.  Aiis.  9. 


MISCELLANEOUS    EXAMPLES.  -O"! 

daughter,  and  the  rest  in  charities  :   how  was  it  appor- 
tioned ? 

18.  A  man  owes  B  $500,  to  be  paid  as  follows  :  SI 50  in 
8  months,  §225  in  6  mouths,  and  $125  in  4  mouths :  if 
paicf  at  one  time,  what  would  the  term  for  payiucnt  be  ? 

19.  A  house  is  50ft.  from  the  ground  to  the  eaves  : 
ivhat  length  of  ladder  would  it  take  to  reach  the  eaves,  if 
the  foot  of  the  ladder  cannot  be  brought  nearer  Ihe  house 
than  30  feet  ? 

20.  If  a  pipe  6in.  in  diameter  will  discharge  a  certain 
quantity  (f  water  in  4  hours,  in  what  time  will  3  four  inch 
pipes  discharge  the  double  quantity  ?         Ana.  6  hours. 

21.  A  gentk^man  left  an  estate  of  $17,500  between  his 
widow  and  son ;  the  son's  share  was  ■§■  of  the  widow's  : 
what  was  the  share  of  each  ? 

22.  What  are  the  prime  factors  of  1400  ? 

23.  What  is  the  hour  when  the  time  past  from  mid- 
night is  equal  to  -^  of  the  time  to  noon  ? 

24.  AVhat  arc  all  the  integral  flictors  of  1100  ? 

25.  How  many  square  feet  in  a  floor  12ft.  Gin.  wide  and 
14ft.  8in.  long? 

26.  What  is  the  greatest  common  measure  of  102-7  aud 
1781 ?  .  .   •        . 

27.  IIow  much  wood  is  there  in  a  pile  4ft.  wide,  2ft. 
9in.  high,  and  IGft.  Gin.  long? 

28.  What  is  the  duty  on  300  bags  of  coffee,  each  weigh- 
ing, gross,  IGOlbs.,  valued  at  7  cents  per  lb.,  2  per  cent, 
being  the  tare  allowed  and  20  per  cent,  the  duty  ? 

29.  What   is   the   square   of   the    cube   root   of  -J-  of 

30.  How  many  bricks  9in.  long,  42in.  wide  and  2in. 
thick,  will  build  a  wall  6ft.  high  and  13Jin.  thick,  round 
a  yard,  each  side  of  the  same  being  280  feet  on  the  out- 
side of  the  wall  ? 

31.  What  is  the  interest  of  §276  for  3  years,  at  7  per 
cent.  ? 

32.  Two  men  speaking  of  their  ages,  one  said  that  f  of 
his  age  equalled  f  of  the  other's,  and  that  the  sum  of  both 
was  90  :  v/hat  were  their  ages  ? 

33.  An  invoice  of  goods  was  sent  from  England  with 
instructions  to  sell  them  and  invest  the  proceeds  in  cotton, 
after  deducting  a  commission  of  li  per  cent,  on  the  sales, 
and  1  on  the  purchase  of  cotton.  The  goods  were  sold  at 
an  advance  of  5  per  cent,  on  the  invoice  price,  and  the 


iH  '     MISCELLANEOUS    EXAMPLES. 

amount  of  §12,600  was  received :  what  was  the  invoice 
price,  and  what  sum  was  invested  in  cotton '/ 

Am.  Invoice,  $12,600;  luvestmeat,  $12,288.1  l-iVr- 

34.  Divide  .$150  among^4  persons,  so  that  when  A  has 
i  dollar  IJ  has  4,  C  i  and  D  i- 

35.  Said  A  to  B,  my  horse  and  saddle  together  arc 
worth  5>I50;  but  my  horse  is  worth  9  times  as  much 
as  the  saddle  :  what  was  the  'value  of  each  V 

36.  A  drover,  with  beeves  and  sheep,  was  asked  how 
many  he  had  of  each  kind.  He  said  there  v. ere  174,  but 
that  the  beeves  were  -j^  of  the  whole  :  how  many  of  each 
was  in  the  drove  ? 

37.  W  hat   is   the   cube   of  the   square   root   of  -f-   of 

•Lf_jof-y? 

4§  ^450 

88.  A  urocer  mixed  !)51bs.  of  sugar  that  was  worth  7 
cents  per  lb.,  65  that  was  worth  D  cents  per  lb.,  and  25 
worth  12  cents  per  lb. :  what  was  the  mixture  worth 
per  lb.  i* 

89.  A  m:in  driving  some  sheep,  cows  and  oxen,  being 
afiked  the  number  of  each  kind,  said  that  he  had  twice  as 
many  sheep  as  cows  and  three  times  as  many  cows  as 
oxen,  and  that  the  whole  was  80:  what  number  of  each 
kind  did  be  have  ? 

40.  A  colonel  forming  his  regiment  into  ahullow  square, 
found  that,  when  he  arranged  the  men  8  deep  on  each 
side  of  the  square,  lie  had  114  men  left;  but  when  he 
arranged  them  4  deep,  he  wanted  114  men  to  complete 
the  arrangement :  what  number  did  his  regiment  consist 
ofy  "  Ans.  750. 

41.  A  merchant  has  due  a  certain  sum  of  money,  of 
which  i  has  to  be  paid  in  2  months,  i  in  3  months,  and  the 
rest  in  6  months :  in  what  time  ought  the  whole  to  be  paid  ? 

42.  A  man  built  a  house  of  4  stories ;  in  the  lower 
story  there  were  16  windows,  each  containing  1:^  panes  of 
glass,  each  pane  16in.  long,  12in.  wide;  in  the  second  and 
third  stories  there  were  18  windows,  each  of  the  same 
size;  in  the  fourth  story  there  were  18  windows,  each 
containing  (i  panes,  18  by  12  :  how  many  square  feet  of 
glass  were  thete  in  the  house  ? 

43.  \Vhat  is  the  difterence  between  3  times  6  and  18, 
and  3  times  18  and  6?  • 

44.  What  number  added  to  the  25th  part  of  2600  will 
make  the  sum  145? 

45.  A  merchant  lias  spices  at  9d.  per  lb.,  at  Is.,  at  2s., 


MISCELLANEOUS  .EXAMPLES.  20r' 

at  38.  per  lb.  :  how  much  of  each  kind   must  he   mix   to 
sell  the  same  at  Is.  fid   per  lb.  ?  ' 

46.  What  is  the  difference  between  G  dozen  dozens  aul 
5  dozen  dffzcns  ?  ^ 

47.  Write  two  millions,  two  hundred  and  two  thousand, 
two  hnn.ived  and  twenty-two. 

48.  What  is  the  value  of  f  of  a  cwt.  ? 

49.  What  is  the  value  of  .325  of  £1  ? 

50.  Reduce  12s.  3d.  2fj[rs.  to  the  decimal  of  a  pornid. 

2.  2« 

51.  What  is  the  difference  between  \/—  and    -5  ? 

lo  lo 

52.  Whnt  is  the  value  of  v/3G0.O0O  ? 

53.  Three  merchants,  A,  13  and  C,  freight  a  ship  v/ith 
tobacco.     A  put  on  board  300  tons,  B  150  and  C  85.     In  '     ' 
a  storm  frherc  was  thrown  overboard  1 25  tons:  what  was 

the  loss  to  them  several Iy1 

54.  How  many  boards  20  feet  long  and  15  inches  v/ide 
will  it  take  f:r  a  floor  40  feet  long  and  30  wide  ? 

55.  A  grocer  bought  a  hogf^heud  of  brandy  for  §87  on 
6m.  credit,  uiid  sold  it  for  cash,  "with  an  advance  on  cost 
of  $18  :  hov/  much  'v^as  his  gain,  allowing  money  to  be 
worth  7  per  cent,  per  an.  ? 

5S.  What  is  the  difference  of  time  between  3Iay  10, 
1860,  and  August  15,  18G5? 

57.  Honv  many  hours  from  October  10, 1860,  at  4  P.  M., 
to  January  1,  1833,  at  7  A.  M.? 

58.  Two  men  hired  a  pastiure  for  $35  ;  A  put  in  3  horses 
for  4m.,  and  B  5  horses  for  3m. :  what  ought  each  to  pay  ? 

59.  A  groc3»  bought  a  hogshead  of  molasses  for  $30, 
but  8  galloiis  having  leaked  out,  he  wished  to  sell  the 
remainder,  so  as  to  gain  4  per  cent,  on  the  whole  cost : 
at  what  price  per  gaUon  must  it  be  sold  ? 

60.  Divide  ^750  between  3  persons,  so  that  the  second 
shall  have  t^  as  much  as  the  first,  aud  the  third  2  as  much 
as  the  other  two. 

61.  A  merchant  sold  a  piece  of  cloth  fn-  §45,  and  lost 
*by  the  sale  8  per  cent. :  what  did  the  cloth  cost  1  • 

62.  A  •eiit^eman  being  asked  the  time,  said  that  the. 
time  past  noon  was  equal  to  -^  of  the  time  past  midnight : 
what  was  the  hour  ? 

63.  What  is. the  discount  of  v?260.85  for  1  yedr  at^d  3 
months,  when  interest  is  8  per  cent.  ?  .    ., 

64.  A  persou  was  born  on  the  Lst  day  December,  1814, 
at  4  o'clock  ill  tjie  uiorning  :  what  v/as  his  age  1st  day 
May,  1864,  at  10  o'clock  in  the  morning? 


-'"<>  MISCELLANEOUS    EXAMPLES. 

65.  What  is  the  area  of  a  square  piece  of  laud,  the 
sides  of  which  are  27  chains?  Ans.  72a.  3r.  24p. 

66.  A  merchant  shipped  14  bales  of  cotton  at  £-^0  lOs 
sterling,   4Hihd.   of  tobacco   at  £14   8s.    3A    per   hhd.,. 
210bbl.   of  rice  at  £3   9s.  6d.   per  b^l. :    what  was  the  « 
amount  in  American  money,  with  exchange  5  percent.? 

67.  A'n  inclined  plane  is  40  feet  long  and  5  feet  high  : 
what  weight  will  be  balanced  by  a  power  of  200  feet? 

Am,  1600. 
6S.  A  lot  of  land,  measuring  39  feet  by  16  J,  cost  $2500 : 
what  was  that  a  square  fobt  ? 

6'J.  What  is  the  area  of  a  parallelogram,  the  height 
being  360  feet  and  width  271  yards? 
.     .'  Ans.  32,520  sq.  yd. 

70.  A  merchant  owes  in  England  £350  15s.  3d.,  but 
has  shipped  15  bags  of  cotton  at  the  valuation  of  $90  each  : 
what  is  the  remaining  indebtedness? 

71.  What  is  the  area  of  a  circle  whose  diameter  is  10  ?• 

72.  AVhat  is  the  commission  on  the  sales  of  250hhd. 
of  sugar  at  $55  per  hhd.,  at  3^  per  cent.,  and  2^  per  cent, 
for  guarantee  of  sales  ?  , 

73.  Th'ere  is  due  in  England,  for  a  bill  of  merchandise, 
£250  sterling:  how  many  dollars  must  be  remitted  to  pay 
for  the  bill,  reckoning  a  £  at  $4.87  ? 

74.  What  is  the  area  of  a  circle  whose  diameter  js  14  ? 

Am.  J6,96S, 

75.  There  are  due  in  Paris,  for  several  pictures, 
4565  francs  :  what  amount  in  dollars  and  cents  must 
be  remitted,  reckoning  19 J  cents  to  the  franc? 

76.  A  note  of  $350,  dated  Octobef  7th,  and  pay- 
able 6  months  from  date,  i.s  to  be  discounted  on  the 
22d  day  October,  at  the  bank  discount  of  7  per  cent. : 
what  must  be  considered  the  worth  of  the  note  on 
the  day  of  discount  ? 

77.  What  is  the  value  of  $1000  of  bank  stock  'at 
105  per  cent.,  or  5  per  cent,  advance  ? 

78.  What  is  the  least  common  multiple  of  2,  7,  14 
hnd  49  ? 

79.  A  merchant  shipped  to  Havana  llObbl.  flour, 
at  $5.75  per  barrel,  and  received  in  return  325  boxes 
of  sugar,  at  $340  per  box  :  what  amount  does  he 
still  ow'e  ? 

80.  The  double  and  the  half  of  a  certain  number, 
increased  bv  2 J,  make  100  :  what  is  the  number? 


APPENDIX. 


BOOK-KEEPING   BY   SINGLE  ENTRY. 

Book-keeping  is  a  plan  adopted  by  business  men  to  Rook-iseepinc 
facilitate  the  interchanges  of  trade,  and  is  simply  an  explained, 
arithmetical  record  of  inorcantile  transactions.     It  is 
of  two  kinds,  single  and  double  entry.     The  latter 
is  used  Avhen  commercial  engagements  are  very  ex-    ' 
tensive  and  great  accuracy  is  desired  ;  but  the  former 
is   most   simple,  and  is  sufficiently  methodical   for 
ordinary  trade. 

Single  Entry  is  the  kind  treated  of  in  this  work. 

The   Day  Book  is  for  all  mercantile  charges  «ind  rpf^^,  ^^^^  i^j^p,. 
credits.     At  short  intervals  these  should  be  posted, 
that  is,  transferred  in  date  and  amount  to  a  book 
called 

li\\Q  Ledger.     This  contains  the  summary  account  ,j,|^^j^^^^^ 
of  each  individual  whose  name  is  in  "the  Day  Book, 
with  numbers  referring  to  the  specific  page  of  trans- 
actions. 

The  Cash  Book  shows  receipts  and  expenditures.     The  cash  book. 

The    Bank   Book   shows    the    deposits   and   with- The  bank  book, 
drawals  of  sums  lodged   in    a  bank  for  safety  and 
convenience  in  transacting  business. 

The  Bills  and  Notes  Pai/ablc  shows  to  whom  and  The  biiis  paya- 
when  an  amount  is  due  ;  and  *''*'■ 

The  Bills  and  Notes  Receivable,  by  whom  a  stated  The  bills recciv- 
sum  is  to  be  paid,  and   the  time  of  expiration  of"*''^- 
credit. 


-08  «0UK-KEK1'1NG. 


REMARKS    ON    NOTES. 

.4;  joint  and  sev     A  j'oint  and  Several  note  can  bo  eoliccted  by  either 
oiraipote.         of  the  signers.  ^ 

'ihe  endorser..     The  endorser  of  note  is. liable  foi-  tLc  anion lU. 
■Anotcnotnago-     A  note  is  uot  negotiable  ^.vhon  the  words  payable 
T.iabie.  I    ^       a  ^q  order"  are  omitted.  • 

Notes  without      All  notcs,  without  the  oxprcfisiou   "  for  value  re- 
value. ■  ccived,"  are  valueless.- 

when'payabie      When    a  uotc  is  paj'oble  to  A  B  or  bearer,  the 
to  bcaror.         t;i<!;ner  is  responsible  to  the  presenter  only. 
Day.s  of  '.vv..       \   iioto  payable   at  a? bank  has  an  extension  of  H 

i]:'.y.<  beyond  its  stated  term.  •  This  is  called  grace. 

When  the  last  of  tkesc  days  happens  on  Sunday  or 

any  recognized  holiday,  payment  nv^  -  ^■"  made  on 

the  day  preceding. 
BUsijiGisuaics:      iS^otcs  passing   commeroially.  till  (111- li    banks   are 
•      jj  either  those  given  in  payment  bf.a.purcbasc,  and 

therefore  called  business  notes,  or  'those  for  which 
Ac>;oii.ivio;ii-  the  bank  has  advanced  money,  called  accommoda- 
iiou paper:  ^j^^^  paper.  Notes  are  often 'lodged  in  banks  by 
oiiection  holders  simply  for  collection.  "When  a  note  remains 
notes:  ,  unpaid  it  is  protested;  or  a  notification  made  to  par- 

xotes  unpaid,  ^:.q^  interested  of  its  non-payment. 


BOOK-KEEPING. 


209 


r(    .       .  :;r  1  A  ].    FO  u  \l  s. 

;;i:GoriAiiLE  notes. 

Charleston,  S.  C,  Feb.  4,  1863. 
For   value    i\ .  ..ived,   I    promise    to   pay   Messrs. 
McCartor  and  Dawson,  or  order,  fi^^e   huiKlreJ  dol- 
lars, on  demaiul,  with  interest. 

WASHINGTON   IRVING. 
5500. 


WANNAH,  Ga.,  June  19,  1865. 
Four  iiioiii;  s  n' i];i  date,  I  promise  to  pay  Ogle- 
thorpe  Hall,   or   order,  three   hundred   dollars,  for 
value  received. 

STEPHEN  DRAYTON. 
$300. 


New  Orleans,  April  5,  1863. 
Sixty  days  from  date,  we  jointly  and  severally 
promise  to  pay  Mr.  F.  Vf.  Pickens  seven  hundred 
dollars,,  for  value  received. 

JAMES  J.  McOAETEB, 
EDMUND  ifAWSON. 
5700. 


FOR.AI    OF    ORDER.  , 

Mobile,  July  4,  I860. 
Messrs  IIayne,  Aiken  &  Co. 

Gentlemen:  Please  pay  to  the  order  of  Tlon.  C  G- 
Memminger  two  thousand  dollars,  and  charge  to 
hcdient  servant, 

JEFFERSON   DAVIS. 

Note. — The  order,  before  it  can   be  paid,  must  be 
endorsed. 


210  BOOK-KEEPING. 


RECEIPTS. 

Columbia,  S.  C,  April  19,  1864. 
Received  of  Ethelwald  Preston  one  hundred  dol- 
lars, on  account. 

WILLIAM    MANNING. 


Knoxville,  Tenn.,  Aug.  9, 1865. 
Received   of  James   Montgomerj^    three   hundred 
dollars,  in  full  of  all  demands. 

JOSEPH  ALDINGTON. 
$300. 


FOKM    OF.  A   BILL. 

Richmond,  Va.,  Jan.  1,  1864. 
Mr.  Henry  Addikgton, 

Bought  of  Palmerston,  Russell  &  Co. 

1  bale  Plains,  500  yards,  @  SOcts $150 

2  bales  Honie?pun,  20  pieces,  600  yards,     @  7cts.     42 

4  dozer;  Blankets,  @  $3  per  pair 72 

4  dozen  Scotch  caps,     @  $4 16 

§280 
Received  Pavmor.t, 
PALMERSTON,  RUSSELL  &  CO. 


BOOK-KEEPINO. 


211 


DAY   BOOK. 


Remarks. 
6m.  cr. 

E. 

January  1,  1862. 

William  "VVoRpswoRxn,              Dr. 

To  5  pes.  Calico,  150  yds.,  @  10c. 

20  pes.  Shirting,  600  yds.,  @,  12'c. 

5  pes,  Osiiabiuf^s,  100  yds.,  @  10c. 

10  pes.  .Sheeting^  200  yds.,  @  '26c 

$15,00 
75,00 
lOiOO, 
50  i  001 

150 

00 

4m.       E. 

5. 

James  Argtle,                             Dr. 

To  10  pes.  Scotch  Plaid,  400  yds.,  @''75c. 

300  00 

Paid   to 
BAG, 

Factors. 

E. 

20. 
William  Wordsworth,            Dr. 
To  Cash  paid  his  order, 

5545 

4m. 

E. 

30. 
William  Wordsworth,              Dr. 
To  100  pes.  Kersej-,  300  yds.,  @  60c. 
50  pes.  Plains,    100  yds.',  (oj,  r.Oc. 
50  pes.  Bro.  Homespun,  2000  yds,  @5c. 
20  pes.  Chintz,  600  yds.,  @  6c. 
1  doz.  Ildkfs. 

150  00 

30  00 
100,00 

36  00 
1,00  817 

00 

E. 

19. 
William  Wordsworth,               Dr. 
To  pd.  his  order  to  Walter  Scott, 

1 

*19 

35 

30  daye. 

E. 

March  9. 
James  Argylk,                             Dr. 
To  2  pes.  Kersey,  100  yds.,  @,  60c. 
20 pes.  Bro.  HoniespuD,  600yds..@5c. 
60  pes.  Long  Cloth,  1200  yds.,  @10c. 

50 

SO 

120 

00 
00 
00 

200 

1 

'bo 

E. 

April  6. 
James  Aegyle,                           Or. 
By  Cash. 

350 

00 

9. 
James  Akgtle,                             Dr. 
To  20  pes.  Calico,  600  yds.,  @,8je. 

50 

00 

■ 

E. 

July  1. 
Wn.i.iiVM  Wordsworth,               Cr. 
By  Cash, 

600 

00 

■ 

E. 

4. 
Casfi  Account,                             Cr. 
Rv  Cash  pd.  travelling  expen. 

360 

00 

3m. 

E. 

21. 

Willlam  Wordsworth,              Dr. 

To  2  cases  Shirtings,  3000  yds.,  @  10c. 

300 

00 

^J2 


BOOK-KKEPiKG. 


1 

Remaeks. 

E. 

July  2t,  1862.  ' 
James  Argyle,                             Cr. 
P>y  order  on  Str.ai-t  &  Co. 

)275'< 

Aug.  5. 
WirXIAM  WOKDSWOUTH,                   Dr. 

To  1  case  plains,  40  p<><?.  I'OOO  yd?.,  ©SOc. 

500|(!' 

L'd^''<Jfor 
Collecfu 
in  Bk.  St. 

E. 

10.' 
Javes  Aegylk,                             Cr. 
By  note  ni  3ni. 

.  1 
,    1 

4in. 

E. 

2(1. 
Jamk^  "Argylk,                           T>v. 
To  10  pes,  Pliiins,  8uu  vtls.,  (^t,  3*0. 
2  doz.  lidkfu.,  @  2| 

90'00         1 
,5!60|    96J50 

Wirxi.VJI    WORDSWOETU,                     <!■. 

By  ^as]j, 

500  0(1 

E. 

,             -     Oct.  9i 
William  WoRoswoR'ni,              Dr. 
y>  1  bale  Blankets,  2<(i  ;  r  ,      ^  1];^ 

on 
00 
00 
00 

400 
600 

00 

C'.n. 

. 

12. 

J.'.MEs  Argyle,                           Di-. 

To  1  cusc:  Ga.  Plains,  lOCO  yds.,  (^ilOi: 

1  case  Bro.  llbnitsp.,  1000  yds.  @  6c. 

1  case  Blankets,  100  pairs,        @.$2. 

20  pe.^  Loii--  Clolh,  400  yds.,    (il  10c. 

300 
60 

200 
40 

Ol' 

~1  <J  V.    t) . 

E.  ]     William  "WoKtswouTU,                Cr. 
il>y  note  iit  3m.                                         ' 

600  00 

E. 

6. 
Jas,  Argyie,                               (  1 . 
By  Cash, 

>i?A, 

41 

Dec.  1. 

Wjt.  WoRiiswon'rn, 

By  bal.  to  new  aecoxmt, 
^- 

Ja8.  Akgtle, 

By  Cash  to  bal. 

Jan.  I,1l863. 

Wm.  Worhsworth, 

To  bal.  from  old  acct, 
^  ^  ^^ 

Wm.  WoRi'SwoKT.ir, 


Cr. 


Cr. 


Dr. 


41  0(1 


'■32'? 


00 


41 10(1 


Dr. 


To  2  bales  Broadcloth,  300  yds.,  @  §2. 


I '600 ;  00 


fi 


Wm.  Wordsworth, 
By  Cash, 


Cr. 


350  0(1 


Noti — The  letter  E  shows  that  the  charge  hnabcen  entered  into  the  Ledgei- 
J^otc. — It  \rill  be  noticed  that  a  few  names  and  entries  only  are  give;. 
These  are  sufficient  to  show  the  method. 


nOOK-KEEPINO. 


213 


Dr. 


LEDGER. 
WILLIAJI  WORDS WORTU. 


i.;.j. 

Prtge !       ■ 

1 

^]  1 

1861. 

Page 

1 
1 1 

.fftn.    1 

1 

'To  am 

tfr.D.ft] 

?l,-0  00 

July  1 

0 

I 
Byani'i  A-.D.hJ'SdoO 

00 

'-I'l 

•  §ept.  5 

11 

''  ,'.0O 

00 

: ) 

f, 

■    " 

:    317,00 

1          1 

Nov.  5 

ir. 

!  coo 

00 

l'(r>.  lii 

.. 

19  3r. 

1 

Dec.  1 

17 

Bal.toaewac't'      !] 

00 

rui.v2i 

0 

300  00 

1 

'1 

\ut;.   f) 

:n 

1 

;>00  00} 

1863. 
an.   1 
10 


1741100 


20  jTo  bal.  I     4)  ooi 

21  "  am  t  fc  D.B.Ij  GUOW  .Jan.  10     21    iByam-tfr.D.B.'l  350^00 


214 


BOOK-KEEPING. 


Dr. 


JAMES  ARGYLEL 


Cr. 


1862. 
Jan.  5 
Mar.  9 
April  9 
July  29 
Aug.  20 
Oct.  VZ 


Pagej 
I 
8     JToam'tfr.D.B 

10     I         •'        '• 

10     \ 

ot.     i 


$300 


200,00 


i862.     Page 

April  6.     15 

I 
July  21      20 


50  00,  Aug.  10 


17550 

95  60' 

600,00 


UVl 


1421  00 


Nov.  0 
Dec.  6 


25 
35 
62 


Byam'tfr.D.B. 

$3.iO 

"        « 

275 

"        " 

137 

.. 

331 

BOOK-KEEPING. 


2\b 


Dr. 


CASH  BOOK. 

CASH. 


Cr. 


1862. 


Jan. 


on 


24 


1  To    Cash 
I  liand  I 

April  6, To  Cash  fr.J.i 

I  Ar2vle 
July  1  ToCushfr.W.j 
Wordsworth 
To  Casli  Stu- 
art A  Co. 
To  CiV'ih  W. 
Wordsworth 
Nov.  6! To    (;ash    J. 

Argvle 
Dec.  6  To    Ca.9h    J. 
I  Argylc 


iiooo'oo 

350  00 


Sept.  5 


1S63. 
Ian.  1 


Tobrtl. 


600 
275 
500 


331141 
327'oO 


38  41 


33S3  43 


Feb.  1 


July  4 


By  pd.  not« 
to  Uk.  S.  C. 
By  pd.  rent 
By  Sund.  for 
house 

By  Trav.Exv 
Cash  Sales 
Bal.    to    new 
a'?c"t 


$550,00 
200  00 

250  OOJ 

350  oo; 

2000 j 00 
38 '41 


3383  43 


Note. — Though  not  inserted  in  our  form,  yet  all  receipts  aud  expendi- 
tures are  to  be  entered  iu  Day  Book. 


516 


BOOK-KEEPING. 


Dr. 


BANK  BOOK. 
BANK  OF  THE  STATE. 


Cr. 


To  deposit 


Jan.  1 

5 

7 

9 

10 

14 

1.5 

17 

■M) 


Feb.  1   To  bal. 


COO j 00 
300  00 


450 
100 
700 

155>)l00 

149  o: 

2C7J33 

I 

•2717  00 


1862. 

Jan.  3 

5 

8 

16 

16 

CO 


451T 


00; 


By  check  to 

"Pnnglo  &  Co, 
Hy  check  to 

Akstuu  &  Co. 
By  ehgck  to 

llnyne  &  Co. 
By  uheekto 

Colcock  &  Co. 
By  check  to 

Cofitin  &  Co. 
By  bill. 


$300  00 

200  00 

400  00 

600  00 

300  OOi 
2717  iOO' 


4517  00  4517  00 


BOOK    KEEPING. 


21^ 


o 

CO 
C5 

30 

CI 

o 
o 

I- 

CO 

1 

• 

. 

CO 

(M 

01 

o 

UO 
CI 

o 

o 

U3 

to 

00 

o 

I- 

i-i 

o  o 
o  -^ 

0 

. 

o 

00 

iC 

1--5 

t— 

1—1 

CO 

CO 

o 

o 

r-t 

I'. 

CO 

h- 

. 

1 

Ci 

o 

<M 

00- 

lO 

l--. 

o 
o 

CO 

o 

o 

o 

1 

•* 

o 

o 
1—1 

CO 

c^                       .                1 

J-t 

S..C).....U...O. 


.;eoOOi-<»Oi-iOOi-i(Mu'5C5'*rt 


218 


BOOK    KEEPING, 


o 

00 


< 

m 
\A 
H 
0 
"A 

Q 
< 

a. 

M 


% 

w 

ca 

< 

M 

o 
o 

1 

c5 

o 

OS 

00 

«^                o 

eo                   o 

I'. 

CI 

■  t  ■              *^ 

■          ,                 ' 

CI 

CI 

CI 

c< 

A       >0 

CI 

>; 

CI 

d 

! 

§ 

'~                                ' 

00                                                                                                                                            1 

1— 

•                                                    lO 
CI 

,    . 

O 

I  g 

1-H 

S 

io 

« 

CI 

• 

^ 

o 

• 

o 

• 

, 

~- 

■  s 

1— 

e^ 

00 

f-l 

1 

l~ 

o 
o 

1 

1 

1 

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Jan.   1. 

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Feb.  5. 
April  G. 

10. 

May  1. 

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ERRATA. 


19th  page,  in  the  1st  line,  hcforc  the  word  "figure,"  insert  "exception  of 
the." 

52d  page,  oth  line  from  top,  after  "40,"  insert  "-f"'  for  "X  ;"  same 
page,  8th  Ex.,  take  out  "  3qr." 

.0.3d  page,  KUli  line  from  top,  after  "1200"  insert  "+"  for  "  X." 

65th  page,  lor  Art.  "  9'J"  read  "105;"  same  page,  17th  Ex.,  for 
"months"  read   ''davs." 

87th  page,  in  27th  and  28th  Examples  take  out  "  25t."  and  "  ,39t." 

92d  page,  r2ih  Ex.,  for  "  7cu.  ft."  read  "7cu.  in." 

114th  page,  9th  Ex.,  take  out  "$18.75  per  bbl"  and  read  "$180.75." 

176th  page,  in  325th  Article,  2d  line,  after  "symbol"  insert  "v/,". 


^    The  attention  ol*  Booksellers  and  Teachers  is  invited  to 

'    Jlritljmctical  Series. 

^ —  ^ 

Prepared  lor  ihe  use  of  th«*  Schools  and  Academies — 
Male  and  Feuialf  —of   tlu-  Conleder^te  States :  to  be 
I  is^ed  as  soon  as  j  /acticable. 

I     1.  Lev(*i;4ts  Piiiiiarv   Aiitlmu'ric. 

I    2.  Lcveivtt.N  Mv]]\\\]   Aiirlmi^'tir. 
i 

•5.  L('\erott.s  ("oiiiiikhi   School  Ariilinietir. 

I    i.  Lcvcrdts  Acadciiiiciil   An'tliiiictic 

^IMILAU  TO  TUK   PRECKDJNG.  HUT  ENLAUGED. 

5.  LcvcrettN  Academical  Algol)ra. 

4 

a.  Leveretts  Plane  (jooinctrv. 


J.  T.  PATES80N.  I.  C.  TUCKO. 

J.  T.  PATBRSdN  k  i:0., 
;itj)0(jraj)l]crs  ;ini)  ^>ttfr-^r55   '0'inttrs, 


H 


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